iV r \ / \ / ^ <^ \ LESSONS ELECTRICITY MACtNETISM. 16226 l'aris. - Imprimerie GAUTHIER-VIIJ.ARS ET Fll-S, qiiai des Rrands-Auifustins ij LESSONS ELECTRICITY MAGNETISM, P. UUHËM, HAROK D Ui\ COiMPLE.ME.NTAIRK DK l'UYSIOUE MATIIEMATIQLK ET DE CHISTALLOORAI'HIE AT THE FACULTY OF SCIENCES OF LILI.E. TOME I. ^f^^v^ ' i>/. () LlîS CORPS COiN DOCTORS TO THE PERMANENT STATE. PARIS, GAUTHIER-VILLARS ET FILS, PRINTERS AND BOOKSELLERS or Ul'UEAU UliS LONGITUDES, DE l'ÉCOLE POL VT EC IIMQ U E, Quai des Grands-Augustins, 55. 1891 ( All rights reserved. ) QC -t.\ INTRODUCTION. In 1811, Poisson inaugurated the theory of electrical phenomena; since that time, the study of the laws of electricity and magnetism has been the work of many great physicists and analysts. Their innumerable works have followed one another without respite during the whole century and in all countries; their discoveries form today one of the most vast scientific collections in the world. The time seems to have come to coordinate the results of so many efforts; to unite in a single bundle these researches conceived according to the most diverse ideas, written in all languages, scattered in all journals. It seems that, if this vast synthesis could be achieved, we would find ourselves in the presence of the most beautiful system of natural philosophy that has ever been created by the human mind. In the present work, we have tried, within the limits of our strength, to draw a first draft of this synthesis. Our work will certainly present many imperfections; but perhaps, in spite of its defects, it will prepare other more completed works; if it is so, we will not have wasted our efforts. What we have proposed to write is an exposition, as unified and as logical as possible, of the theories of electricity and magnetism, and not a compilation of these theories. One will not find here all that has been said on electric and magnetic phenomena; we only wish that one finds there the really clear and fruitful ideas which were emitted on their subject. IMKOnUCTION. jet. The ore that contains Science always contains gangue; we have rejected a lot of this gangue: the liter of what we have kept will only be richer. After having meditated for ten years on the various parts of electrical science, we became convinced that all that is clear and fruitful in this science could be grouped, with much order and unity, around a few principles borrowed from Mechanics and Thermodynamics, and it is this grouping that we have tried to expose. Before showing how Mechanics and Thermodynamics allow us to sort out, to sort out, to classify the theories relative to Electricity and Magnetism, should we take up and discuss the principles of these sciences? Logically, yes; for the exposition that is usually given of these principles is sufficient for the examination of some of the simplest problems of Physics; but one finds it, on the contrary, constantly defective and contradictory when one wants to apply it to the more complex questions raised by the study of Electricity and Magnetism. However, this logically necessary revision of the principles of Mechanics and Thermodynamics, we have not done it here. The reason is simple: in order to foresee all the cases, so complicated, that the electrical theories will offer, it is necessary to overload the demonstrations of JVlécanique and Thermodynamique with a lot of restrictions, conditions, hypotheses, which make them one of the most difficult parts of Science. To present these theories in an abstract way, separating them from the applications which require and, therefore, justify this luxury of precautions, would be to make them difficult to understand perhaps, and certainly off-putting. We therefore thought it would be a good idea to reverse the logical order: in the present book, we propose to make the thermodynamic instrument work before the eyes of the reader; to make him carry out the work for which it has been necessary to complicate its workings so much. We will limit ourselves to giving, from time to time, some brief indications on its mechanism. Later, if our strength does not betray us, we will dismantle this instrument piece by piece, we will explain all its INTRODUCTION. Vil organs and we will show the other applications that can be made. This first volume is devoted to the study of the laws governing the equilibrium of electricity and its permanent motion on conducting bodies. After having given an overview of the classical theory of electrical equilibrium and of the mathematical methods used to solve the problem of electrostatics, we elude, in the last three Books, the applications of Thermodynamics to homogeneous or heterogeneous metallic conductors and to electroless conductors. We are very brief on electrolytes, not that the methods employed for metallic conductors do not allow us to go very far in the study of them; but this study cannot be made without a thorough examination of the properties of saline dissolutions, and the length of this examination would exceed the limits of our Work. On the contrary, we will treat the properties of metallic conductors very completely; the exposition that we give is, we believe, the most extensive and the most thorough that has been given so far. The second volume will be devoted to the properties of magnets and dielectric bodies; the methods indicated in our new Theory of the Magnet (ition by Influence, based on Thermodynamics, have allowed us to bring back to unity the vast body of research to which these bodies have given rise, to rectify them on several points, to com[)lerate them on many. In a third volume, we will study the phenomena produced by linear currents, the laws of induction between such currents, their electrodynamic actions, the induction that magnets generate in metal wires, the magnetization by currents, the actions that are exerted between magnets and currents. The new method followed in this part of our work will allow us to study the properties of any linear currents and not only of uniform currents, to which we are generally limited. The method followed in this third Volume will be extended later to the study of the propagation of electricity in INTRODUCTION. media of finite extent in any dimension, whether they are conductive, magnetic or dielectric. We have left aside, in general, the study of the methods for measuring the various electrical quantities; the classical treatises contain sufficient information on these methods. Similarly, we have limited ourselves to a summary of the experiments which serve to verify the laws we are studying; the detailed description of the instruments and the precautions required by their use can be found in other books. Our aim, moreover, is not to write a manual to serve as a guide for the experimenter and the practitioner, but to clearly show the theoretical link between the various parts of electrical science. The clear view of this link will provoke the discovery of new phenomena, of new laws, which, in their turn, will serve to strengthen and extend it. [.ille, I" July 1891. LESSONS SLR ELECTRICITY MAGNETISM. TOME I. BOOK I. ELECTROSTATIC FORCES AND THE POTENTIAL FUNCTION. CHAPTER ONE. FIRST DEFINITIONS. - COULOMB'S LAWS. § 1 - First definitions. Symmer and Franklin brought the first contingent to the set of hypotheses which represent the properties of electrified bodies. These hypotheses were, in their origin, intimately connected with suppositions on the nature of electricity; but it is easy today to break this link, to leave aside these suppositions on the nature of electricity, suppositions so foreign to the true object of Physics, that this science has not even the right to show their vanity; to leave, finally, to the fundamental hypotheses only the character of definitions of analytical parameters which is essentially theirs. 1). - I. I 2 BOOK I. - ELECTROSTATIC FORCES. An electrified system will be composed, for us, of a certain number of bodies, either homogeneous, or endowed with a constitution that defines certain parameters and certain geometric quantities that vary from one point to another in a continuous manner. These bodies are separated from each other and from the non-electrifiable medium that surrounds them by surfaces of discontinuity. To each point M(x,y,z) inside one of these bodies will correspond a quantity p, function of x, y, z, adding in any point M positive or negative values, but finite and that we will name the solid electric density at the point M. If, around the point M, we trace an element of volume di' = dx dy dz, the quantity dq = p dv will be the quantity of electricity, the electric charge, or the electric mass that this element contains. To each point P situated on one of the surfaces of discontinuity which the system contains will correspond a quantity o-, function of the two parameters u and v which determine the position of a point on this surface. This function will have, at any point P, positive or negative values, but finite; it will be called the superficial electric density at point P. If, around the point P, on the considered surface, we trace a superficial element c^S, the quantity dq = TdS will be the quantity of electricity, the electric charge, or the electric mass qvie contains this surface element. The complete set r r r g dx dy dz, Ifl^ extended to the entire volume of one of the bodies that form the system, represents the total amount of electricity spread within this body. ^rale has d'S, The complete set S' extended to all the elements of one of the discontinuity surfaces of the system, represents the total amount of electricity spread over this surface. If we form for all the bodies of the system the integrals analogous to the one we just considered first, for all the discontinuity surfaces of the system the integrals ana CHAP. I. - COULOMB'S LAWS. If we add together all the integrals thus obtained, the sum will represent the total quantity of electricity contained in the system. We will admit that this quantity remains invariant in any modification of the system. This hypothesis was introduced into Physics by Franklin, when he admitted that any electrical phenomenon could be represented by the formation of equal quantities of positive and negative fluid at the expense of the neutral fluid. Since Franklin's time, physicists have not ceased to admit it, explicitly or implicitly, and to make frequent use of it in their theories. Thus, in a Memoir by Ohm, published in 182^, a Memoir whose importance we shall soon have to point out, we find this principle very explicitly stated (' ). "It is a condition," says Ohm, in the course of one of his reasonings, "which results from this fundamental principle, that the two electricities always develop in the same time in equal quantities. "Nowadays, this hypothesis has been given the title of Principle of the Conservation of Electricity (2). § 2 - Dufay's and Coulomb's laws. Once these definitions have been established, we will admit, on the basis of the well-known experiments of Dufay, Cavendish and Covdomb, that if two infinitesimally small material masses m and m' are placed in the presence of each other, at a sensible distance, carrying electricity spread inside them or distributed on their surface, each of these two masses is subject to a force verifying the following laws: 1" The action F' that the mass m exerts on the mass m' is directed (' ) G.-S. Ohm, Mathematical Theory of Electric Currents, translated by J.-M. Gaugain, p. 109. (') G. LiPPMANN, Principle of the Conservation of Electricity or Second Principle of the Theory of Electrical Phenomena {Journal de Physique pure et appliquée, 2" série, t. X, p. 38i; 1881). 4 BOOK I. - ELECTROSTATIC FORCES. along the line that joins a point M of mass m {Jig. i) to a point M' of mass m' . 2" If one designates by q the total electric charge carried by the mass m, by q' the total electric charge carried by the mass -^yl vg/ m', by r the distance MM', by e a certain positive and absolutely constant coefficient; if, moreover, one supposes the force F' counted positively when it is repulsive, i.e. directed from M to M', and negatively when it is attractive, i.e. directed from M' to M, the force F' is represented in magnitude and in sign by the formula F': 99 3" The force F that mass m' exerts on mass m is equal and directly opposite to the force F' that mass m exerts on mass m! We see, by the preceding formula, that, the coefficient £ being positive, the force F' has the same sign as the product qq'; in other words, that two electrified material masses attract or repel each other according to whether they are charged with electricities of different signs or with the same electricity. The value of the coefficient s being absolutely constant, one can, if one wishes, give to this coefficient the value i and, consequently, make it disappear from the formulas, on the only condition of choosing in an appropriate way the unit of electric charge left, until now, arbitrary. If we define the quantity of electricity which must serve as a unit by the condition that two infinitely small material masses, each charged with this quantity of electricity and placed at a distance of one unit from the other, exert a repulsive action on each other equal to the unit of force; if we suppose, moreover, that the charges q and q' are related to this unit, the preceding formula can be written ri CHAP. I. - COULOMB'S LAWS. 3 The simplification which results from this suppression of the coefficient s is small enough to be of little interest in theory. On the contrary, when we deal with the relations between the different systems of electrical units, it will be convenient for us to consider the formulas in which the unit of quantity of electricity has been left arbitrary and in which, consequently, e has been retained. We will therefore leave this quantity in our formulas. Experience has established the accuracy of Coulomb's laws only for electrified masses whose distance is greater than a certain limit. Are the forces that act between electrified masses located at insensitive distances from each other still given by these laws? This is a question that we will examine later (' ). For the moment, we will study the forces that act between electrified bodies, assuming that these forces are exactly and in all conditions given by Coulomb's laws. The results obtained in this way will serve us for the later discussion of Coulomb's laws (-). (-) See Book IV, Chapter II. (") Coulomb's works on Electricity and Magnetism were published, from 1785 to 1789, in the Mémoires de l'Académie Royale des Sciences de Paris. Those which relate to the demonsLralion of Coulomb's laws are the following: First Memoir on Electricity and Magnetism, by M. Coulomb. Construction and use of an electric balance based on the property that metal wires have of having a torsional reaction force proportional to the angle of torsion (Memoirs of the Academy for 1785, p. 069-577). Second Mémoire sur l'Électricité et le Magnétisme, in which it is determined according to which laws the magnetic fluid, as well as the electric fluid, act either by repulsion or by attraction, by M. Coulomb {Mémoires de l'Académie pour 1785, p. 578-611). Coulomb's Memoirs form Volume I of the Memoirs of Physics reprinted by the French Physical Society. BOOK I. - ELECTROSTATIC FORCES. CHAPTER I DEFINITION OF THE POTENTIAL FUNCTION. - PROPERTIES OF THIS FUNCTION AT A POINT OUTSIDE THE ACTING CHARGES. Coulomb's law has a close analogy with the law of universal attraction. Thus, it has been possible to extend to the study of electricity the analytical methods that astronomers had created to solve the problems of celestial mechanics. These methods, transported into a new field, have shown an extraordinary fruitfulness, which has been manifested both by magnificent mathematical theories and by physical consequences of great importance. It is to the properties of a function whose name at least has penetrated everywhere, the potential function, that the consequences we are dealing with are linked. We shall see how Coulomb's laws lead us to the definition of the potential function. Let us suppose first of all, to simplify our presentation, that we replace by points the solid or superficial elements of the electrified bodies and that we concentrate in each of these points the total electric charge distributed on the element to which it corresponds. Let M be a point of coordinates x, y, z, enclosing a quantum Fiir. 2. (j;,y,y^.) /Y tity equal to the unit of positive electricity. Let A, A', A", ... be any number of other points {fig- 2). The point A gives CHAI". II. - THE POTENTIAL FUNCTION OUT OF THE CHARGES. 7 closes a quantity q of electricity, the point A' contains a quantity ^', the point A" a quantity ^", .... The coordinates of point A are a, 6, c; those of point A' are "', Z>', c'; those of point A" are a", h% c" ; Point A exerts on point M an action F, directed from A to M, ([ui has the value /- designating the AM distance. The point A' exerts in the same way at the point M an action F', directed from A' to M, which has the value i' denoting the distance A'M; .... Let R be the resultant action exerted at point M by points A, A', A", . ". ; let X, Y, Z be the components of this action along the three coordinate axes. Let i, r,, î^ be the components of the force F; ^', r/, "Ç the components of the force F'; ^", r/', "Q' the components of the force F"; .... We will have ■ Z = Ç ^-^'-+-C-^-..-. Let us denote by a, p, y the angles that the AM direction makes with the three coordinate axes. We will have ^ = Fcosa, 7) = Fcosp, !^ = F cosy. But, on the other hand, X - a " y - b z - c COSa = j COSp=:-:^ , COSY = ■ - - J /- ' r ' r well dr " dr dr cosa = T-J cosp = -- > cosY = -\ ox oy ' oz according to the equality ri = {x - ay ^{y - by-+{z - cy. 8 BOOK I. - THE FORCES ELECTROSTATICS. We have so l = ^^. r^ dr dx ■') _ - 1 dr r = S^ dr which i can to be eradicated >- = -zg r dx ' ^- - = --g 4r' K- ^-tq d'r We have and also ^- = -zq' r dx -'î ■ = -tq à-, From these formulas, we deduce the following X = J to' r dx + q' r dx -h 9" r dx Y= - J dl r + 7' d^, r dy H 9" to-" r 'W Z = - .J dl r -\ 9' to-, r 'd^ + q" dz or good, in posing (0 V r .9 r r -+ + .. ., CHAP. II. - THE POTENTIAL FUNCTION OUT OF THE CHARGES. 9 and, noting that q^ q\ q" ^ . . . are independent quantities of x^ y, Zj X =--s -7-, The quantity V, whose partial derivatives thus make known the components of the action R exerted at the point M, charged with one unit of electricity, by the charges q, q', q", . . . placed at points A, A', A", . . . . . is called the potential function at point M of the charges distributed at points A, A', A", .... By means of the potential function, we can know the component of the force R in any direction. Let a, b, c be the angles that this direction makes with the three coordinate axes and T the component of the force R along this direction. If we denote by /, m, n the angles that the direction of the force R makes with the coordinate axes and by G the angle that the direction of the force R makes with the given direction, we will have CCS 6 = cosa cosl -\~ cosb ces m -+- cosc cos/i and T = R cos6 = R cosl cosa -+- R cosm cosb -r- R cosn cosc. But X = Rcos/, Y = Rcosm, Z = Rcosn. We will therefore have T = X cosa -r- Y cosô -f- Z cosc. Let us lead by the point M a parallel to the considered direction. Let us designate by s the distance of the point M to a fixed origin taken on this line, this distance being counted positively if, to go from this origin to the point M, we walk in the considered direction. Let M' be another point, taken on the same line, at a distance (s -h ds) from the origin. The potential function has, at this point M', a value V that we will represent by (V-h -r- ds). The quantity -T- thus defined is what we will call the derivative of the potential function along the direction s. lO BOOK I. - ELECTROSTATIC FORCES. / dx The coordinates of point M' will have the values (x-{- ,~ dsV we will have dY _ dV dx dV dy d\ dz ~ds dx ds dy ds dz ds But, on the other hand, dx , dv dz cosa = -,- > CCS 6 = - ,- j cosc= -.- > "5 aces ds so that, if, in the expression of T, we replace X, Y, Z by their values (2), we will have _ /dW dx toY dy , dY dz \àx ds dy ds ~^ dz ds or We have assumed, in the foregoing, that each of the elements of volume or surface contained in the system was replaced by a point of this element, and that the load carried by this element was concentrated in this point. This operation is always permitted, provided that the elements to which the points A, A', A", . . . belong are all at a finite distance from the point M; or, in other words, that one can trace around the point M a closed surface whose points are all at a finite distance from M and which contains no other electricity than the equal charge of the unit which is at the point M. If these conditions are fulfilled, we shall say abbreviatively that the point M is external to the acting charges. Under these conditions, it is easy to see that the expression of the potential function at point M can be obtained as follows: For each of the electrified volumes in the system, let us form the triple integral fil da dh de, p being the solid electric density at a point of the element da dh of, and r the distance from this point to the point M [x, y, z). OHAP. II. - THE POTENTIAL FUNCTION OUT OF THE CHARGES. II For each of the electrified surfaces in the system, let us form the double integral y.dS, 7 being the surface density at a point of the element dS, and r the distance from this point to the point M (x, y, z). Let's add together all the integrals thus obtained, and we will have the value of the potential function at point M (4) \{x,y,z)=^JJJUadbdc+^^~dS. The function V thus defined is a function of the coordinates x^ y, z of the point M. Since the point M remains at a finite distance from all the electrified points, r is always different from o. The element subjected to integration is always finite; it uniformly and continuously varies with the coordinates x, y, z of the point M. The quantity V is therefore a uniform, finite and continuous function of the coordinates x, y, z of the point M. When one of the three variables x, y, z grows beyond any limit, all the quantities r grow beyond any limit; all the elements under the integration sign tend to o, and so does the function V. But the products ^V, J'Y, ^V keep in general, in c^cas, finite values. We have obviously, by virtue of equality (4), J^w dy't dzP ~ jk^J J J ^ dx"^ dy^dzP 28 ()/n+n+p _ + i) of the same variables remains, in general, finite. The relation (6) r^=(x - a)'i-i-(_y - by-h(z - cy gives and, therefore, r i 3{cp - ay Equality (5), which is applicable whenever the point M is external to the acting masses, then gives ôx \=-^JJJ^,dadbdc^Z^j'ff^S^^dadbdc ^^ ^c , Q VC "^i^ - aY ■ss^^-^ss^ ^s. We also have dS, If we add member by member the three equalities thus obtained, taking into account equality (6), we arrive at the following proposition, the importance of which we shall see at every moment: At any point M, outside the space containing the agent, we have the equality ()2V ()2V t)2V _ CHAP. II. - THE POTENTIAL FOXCTION OUTSIDE THE CHARGES. l3 The expression t)2V d^\ ù'^Y dx^ dy^ dz^ occurs so frequently in calculations that it is useful to represent it by a symbol; the symbol AV is generally adopted today in French and German treaties. The preceding equation then becomes {-; bis) . AV=o. We have just briefly sketched the main properties of the potential function at a point outside the space containing the agent. These properties are so important that it would be unforgivable to ignore the history of their discovery (' ). Lagrange, the first, in his General remarks on the motion of several bodies which attract each other in inverse proportion to the squares of the distances (^), pointed out the fundamental property of the potential function: Let," he says, "M, M', M", ... be the masses of the bodies which compose the given system ; x, y, z the rectangular coordinates of the orbit of the body M in space ; x' , y', z' those of the orbit of the body M', etc." 5 that we make, to abbreviate, it = MM' \^{x-x')^ + {y-yy^-i-(z ~z'Y MM" x/{x - x")^-^ {y-yy -t- i-s ■ -z "y^ M' M" and that we usually denote by -t-> - - - the coefficients of dx.^ . . . , in the differential of the quantity Q considered as a function of the variables x^ y, z, x' ... ; we will have I of I dQ M 55' M 5k' I dQ M dz (' ) See Max Bacharach, Abriss der Geschichte der Potentialtheorie (Inaugural dissertation). Wurzburg; i833. (") Read at the Academy of Berlin on October 20, 1777 {^New Memoirs of the Academy of Berlin, year 1777, published in 1779, pp. i55-i74). l4 BOOK I. - ELECTROSTATIC FORCES. for the forces with which the body M is attracted by the other bodies M', M", . . . according to the directions of the coordinates 0", ^, z, . . . This way of representing the forces is, as can be seen, extremely convenient by its simplicity and generality. " Lagrange adds in conclusion (p. l'ji): "One would prove by the same principles that these theorems would also be true if the bodies acted on each other by a force of mutual attraction proportional to any function of the distance. )) Lagrange's remark seems not to have been noticed by his contemporaries, and Laplace must have found the potential function by his own efforts; for the first person to make use of it, after Lagrange, was Legendre (' ), and he refers to the fundamental property of this function as "a theorem that M. Laplace has kindly communicated to me". Laplace used the potential function for the first time in 1784, in his Theory of the motion and elliptical figure of the planets. He then made frequent use of it in his Celestial Mechanics and, by doing so, contributed more than any other to revealing the importance of this function. In the Memoir (2) in which Poisson founded the theoretical Electrostatics, the potential function was, for the first time, used in the research of Physics; the role that it plays there has not ceased to grow since that time. The name of the potential function was given to it by Green (^) in 1828. Gauss ('*) gave it the name of potential in 1840; this name lends itself to some confusion; so Clausius (^) proposed to (') Legendre, Recherches sur l'attraction des sphéroïdes homogènes {Mémoires des Savants étrangers, pp. [\ii-l\i. Paris; 1785). (^) Poisson, Mémoire sur la distribution de l'électricité à la surface des corps conducteurs, lu à rAcadcmie des Sciences les 9 mai et 3 août 181 2 {Mémoires des Savants étrangers, p. i; 181 1). (^) G. Green, An essay on the application of mathematical analysis on the theories of electricity and magnetism. Nottingham; 1828. {*) G. -F. Gauss, Allgemeine Lehrsàtze in Beziehung auf die im verkehrten Verhàltnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungs Kràfte {Mémoires de Goettingue, i84o, Gauss, Werke, t. V). (^) Clausius, The potential function and the potential; translated into French by Folie. Paris; 1870. CHAP. II. - THE POTENTIAL FUNCTION OUTSIDE THE CHARGES. l5 to take up Green's name. It is to be regretted that this example, to which we shall conform in the present work, has not been followed by all physicists. The AV symbol, adopted today in France and Germany to represent the expression d^-Y d^Y d-^Y is due to Murphj ('). Lamé designated this expression by AjV, Betti by A-V, Green by 5V. English authors like to use the symbols VV and V- V. The fundamental equality AV=o is due to Laplace. It is generally called Laplace's equation. Laplace gave it, in this form, only in '7^7 (")- But, as early as ^Sa (^), he had given the equation, transformed from this one into polar coordinates, which serves as a starting point for Laplace's theory of Y" functions. (') MuRPiiY, Elementary principles of the theory of electricity , heat and moleculars attractions. Part I, p. \!{0', i853. (^) Laplace, Mémoire sur la théorie de l'anneau de Saturne (Memoirs of the Academy of Sciences for the year 1787, pp. 249-267. Paris; 1789). (') Laplace, Théorie des attractions des sphéroïdes et de la figure des planètes {^Mémoires de l'Académie des Sciences pour l'année 1782. Paris; 1785). l6 BOOK 1 - ELECTROSTATIC FORCES. CHAPTER m. GREEN'S THEOREM. To further study Ja potential function, we will often need to make use of an analytic identity whose statement is called Green's theorem. To prove this equality, we will rely on a lemma that we will first establish. Consider a closed space bounded by a simply connected or multiply connected surface. Let U and F be two functions of x^y^ z, which are finite, uniform and continuous at all points of this space and of the surface which bounds it, and whose partial derivatives with respect to x are finite at all points of the same space and of the surface which bounds it. Let's consider the integral dF -m^ . dxdydz, dx -^ extended to this space. We will, by means of an integration by parts, transform this integral. Let M be a point on the surface which bounds the considered space. Let us lead a normal to the surface towards the interior of this space. Let us denote by (N/, x) the angle formed by this normal thus directed with the positive direction of the x axis. Let us lead, at point M, a parallel to the positive direction of the X axis. If, at point M, this parallel penetrates inside the considered space, the angle (N/, x) will be acute. It will be obtuse if this parallel leaves the space. Let us lead a plane perpendicular to the x-axis^ leaving the considered space entirely on the side of the positive x's. All the points of this space project on this plane inside a certain closed curve, with single or multiple connections, which is the apparent contour of this space. Let us divide the area inside this closed curve into surface elements. CHAP. III. - GREEN'S THEOREM. I7 Let f/S be one of these elements. Through all the points of the contour of the element dH, let us lead parallels to the x axis. We will thus obtain an infinitely untied cylinder which will cut on the surface of the considered space an even number of elements. We will designate these elements, according to the order in which they are encountered when we advance parallel to the ce axis, by the indices 1 , 1, . . . in. Their surfaces will be dSt , dSi, . . . dS2n- At a point of an element of odd rank, a parallel to the positive direction of the x-axis penetrates inside the space; the angle (N/, x) is acute. On the contrary, at a point of an element of even rank, a parallel to the positive direction of the x-axis goes out of the enclosed space; the angle (N/, x) is obtuse. We can therefore write (I) dl. = dSi cos(N/, x)i r= - dSz cos(N/, x\ = dS$cos{Ni,x)i =... = _ dS2nCos{N,; a;),^. Let Xi be the abscissa of a point of the element c^S, ; x^ the abscissa of a point of the element dS2, ... ; Xin the abscissa of a point of the element dSo/i' We will obviously have s-U \ràF , r""^ dF , r d¥ l] - dx+ l f {]-^-dx+.. ,.-i- / \]~dx dx J at .L âx the sign V indicating a summation that extends to all elements -^-^ are assumed finite at all points of this space and of the surface which limits it. CHAP. III. - GREEN'S THEOREM. I9 Let's use equality (2), assuming that we have In the same way ///^ ^ ^-^ ^^ ^" = - s ^ s ''^'^^'' "^ ^^ ""/// s rz'^"' '^^ '^" Let's add member to member the three equalities thus obtained and we will find Ç Ç] b.y dx dy dz = - S U [^"os(N/, ^) + ^cos(N,, j) + ^ cos(N.-, ^)] d^ J J J \dx dx dy dz dz ) "^ "^ If we denote by -r^ the derivative of the function V along the normal to the surface which limits the considered space, this normal being directed towards the interior of this space, the very definition of the derivative of a function along a direction will give us and the previous equality will become ( f C Ç\] t,Y dx dy dz This is the identity that is known as Green's theorem. This identity of Green is often given another form. 20 BOOK I. - ELECTROSTATIC FORCES. Let us assume that the first order partial derivatives of U are uniform, finite and continuous, like those of V, inside the space under consideration and on the surface which bounds it, and that the second derivatives -r-^ > -^-r > -r-r exist and are limited in all OX^ 0/2 ()z^ points of this space and of the surface which limits it. We can write r f Cy M] dx dy dz J J J \àoc dx dy dz dz j "^ kj dN, Comparing this equality with equality (3), we find This new form of Green's theorem is often used to transform a triple integral extended to a closed space into a double integral extended to the surface which limits this volume, a transformation which one has at every moment the opportunity to carry out in Mathematical Physics. By means of particular assumptions made on the functions U and V, one can deduce from formulas (3) and (4) other often used formulas. If, for example, we assume U = V, the identity (3) becomes Ç C Çy t,y dx dy dz If we make 11 = 1, either in identity (3) or in identity (4), we find (6) JJJ^Ydxdydz^-^^dS. In the rest of these lessons, we will often have to make use of equalities (5) and (6). Green was the first (' ) to insist on the great importance of (' ) Georges Green, Essay on the application of mathematical analysis on the theories of electricity and magnetism (Nottingham, 1828). (5) CHAP. III. - GREEN'S THEOREM. The formula (4), says M. E. Mathieu (' ), is the one that serves to determine the coefficients of the series that gives the cooling of a body; it was therefore used in various cases by Fourier and Poisson, long before the appearance of Green's Memoir on the theory of Electricity. "The lemma which we have used to establish it was often employed by Poisson. (-) Emile Mathieu, Théorie du potentiel et ses applications à l'électrostatique et au magnétisme. First Part: Theory of Potential, p. i4 (Paris, i885). ELECTROSTATIC FORCES. CHAPTER IV. GAUSS'S LEMMAS. ATTRACTION OF A HOMOGENEOUS SPHERICAL LAYER. § 1 - The three lemmas of Gauss. The demonstration of Green's identity is essentially based on the division of a closed space into an infinite number of infinitely unbound cylinders having all their generators parallel to each other. There is another very natural way of dividing a similar space into infinitely small volumes: it consists in decomposing this space into infinitely sharp cones all having their vertex at the same point. This mode of division was first used by Lagrange in the study of the problem of the attraction of ellipsoids (' ). Legendre found, for his part, this same mode of decomposition of space in his first work on the attraction of ellipsoids (2). In his second Memoir on the same problem, Legendre (^) recognizes, in this respect, the priority of Lagrange: "This principle," he says, "to which I had arrived by geometrical considerations, was not found to be different from a means of transformation indicated by M. de Lagrange... The property thus belongs to this illustrious geometer. " This method of dividing space proved to be very powerful in the hands of geometers who dealt with the theory of attraction. In particular, it allowed them to solve the problem of the attraction of ellipsoids. There is a kind of parallelism between the methods based on the decomposition of the space into infinitely untied cylinders whose generators have the same direction and the methods based on (') Lagrange, Mémoires de l'Académie de Berlin, p. i25; 1773. (') Legendre, Mémoires des Savants étrangers, t. X; 1778. (') Legendre, Mémoires sur les intégrales doubles {^Mémoires de l'Académie des Sciences, p. 4^5; 1788). CIIAP. IV. - LEMMAS OF GAUSS. 23 on the decomposition of space into infinitely untied cones originating from the same point. This parallelism has been pointed out several times by Gauss (' ). The role played, in the first kind of methods, by Green's theorem, seems to be held, in the second kind of methods, by three very simple lemmas that Gauss (^) has imagined in order to solve the problem of the attraction of ellipsoids. It is these lemmas that we will demonstrate, following Gauss' own form. Let us imagine a closed surface S, with single or multiple connections, bounding a certain closed space E; let us also imagine a point M which, for the moment, we will suppose to be not located on the surface S. From point M, let's start a ray vector that meets the surface S; if point M is inside the space E, this ray vector will meet the surface S an odd number of times. At each encounter of odd order, it will leave the space considered, and at each encounter of even order, it will enter it. If, on the contrary, the point M is outside the considered space, the ray vector will meet the surface S an even number of times; at each encounter of odd order, it will enter the considered space; at each encounter of even order, it will leave it. Let us denote by /-^ the distance from point M to the yo'th point of encounter of our ray vector with surface S. Let N^ be the normal at this point to the surface S, this normal being directed towards the outside of space E, and let (/\Ne)p be the angle that the normal thus directed makes with the ray vector. The angle (r,^e)p is acute or obtuse, depending on whether the ray vector exits or enters the space E. Therefore, if the point M is inside the space E, we have ( ' ) Gauss (C.-F.) , Tlieoria attractionis corporum sphœroidicorum ellipticorum homogeneorum methodo nova tractata {New Memoirs of Goettingue, t. II, i8i3; Gauss, Werke, t. V, p. i). - Principia generalia theoriœ figurœ Jîuidoruni in statu œquilibrii {New Memoirs of Goettingen, t. V, i83o; Gauss, Werke, t. V, p. 43). (") Gauss (G.-F.), Theoria attractionis corporum sphœroidicorum ellipticorum homogeneorum methodo nova tractata {^Nouveaux Mémoires de Goettingue, t. II, i8i3; Gauss, Werke, t. V, p. 9). 24 BOOK I. - ELECTROSTATIC FORCES. rons cos(r, Ne)i>o, cos(r, Ne)2 o, and, if the point M is outside the space E, we will have cos(;-, Ne), o, ..., cos(r, Ne)2"> o. On the surface of a sphere of rajon i, having for center the point M, let us take an element of y close to the point where this spherical surface is pierced by our rajon vector. The contour of this element being taken as the director of an infinitely loose cone whose vertex is at M, this cone will cut out on the surface S elements CtS \ A CiS 2} - "5 as py - - f . If the angle (r, N^)^ is acute, we have _ COs(/-, l^e)i>d^p _ ri If, on the contrary, the angle (/-, N^)^ is obtuse, we will have ^(j = - COS(r. 'Ne)pdSp ' p This being said, let us consider, first of all, the case where the point M is inside the space considered. We will have _cos(r, Ne)i and, therefore, cos(r, Ne)i "iSi cos(/-, Ne)2<^S2 , cos(r, Ne)2/"-i c?S2,"_i , ■ 11 ' -^ -T- . . . H -^ =: rfa. ' l '2 ' 2'n-l Let us write the analogous equalities which are given to us by all the elements "?o- of the sphere of radius i, and let us add them member by member; we will see without difficulty that the first member of the resulting equality will be the integral S COS(/-, Ne)ndSzn '211 and, therefore, cos(r, Ne)ic?Si cos(r, Ne)2 Let us add member by member the analogous equalities provided by all the elements of s of the second hemisphere, and we will find (6) g cos(r,N.)rfS ^^^ CHAP. IV. - LEMMAS OF G.VUSS. 27 the integration extending to all the elements of the surface S which are in the second region. Let us add member by member the two equalities (a) and (6); the sum of the first members will be the integral Scos(r, Ne)rfS extended to the whole surface S, and we will obtain the third Gauss lemma: When the point M is located on the surface S and the surface has a unique tangent plane at this point, we have cos(r, Ne)c?S (3) S^'<^ If at point M the surface S, instead of admitting a single tangent plane, admitted several tangent planes or a cone of tangents to one or more sheets, a similar reasoning would easily give the value of the integral Scos(r, Ne)ifS If, for example, the point considered is a simple conic point, we have a being the spherical opening of the sheet of the cone which encloses the surface in its interior in the vicinity of the point considered; if the point considered is part of an edge, the same formula applies, - being the plane angle of the dihedral which, in the vicinity of the edge, includes the surface inside it. § 2 - Existence of the force of attraction at a point inside the acting charges. Consequence of Gauss' lemmas. Let us suppose that a system is formed by one or more bodies containing electricity in their interior whose solid density is p at the point (", ^, c). This system contains surfaces of discontinuity on which electricity is distributed, and its surface density is a at a point of the surface element of S. Let M(.2;,y, c) 28 BOOK I. - ELECTROSTATIC FORCES. a point outside all these acting masses. If this point carries a unit of electricity, it will be subjected, by virtue of Coulomb's laws, to a force whose components are /' designating, depending on the circumstances, either the distance from point M to point (a, Z), c), or the distance from point M to a point of the element dS. Let us now take the point M(a:,y, z) k inside the regions occupied by the acting charges, either at a finite distance from any surface of discontinuity, or even on a surface of discontinuity. Let us leave aside, for the moment, this last case, which is the most complicated. Let us designate by the index i the electrified body to which the point M belongs, i.e. a mass of finite extent, surrounding the point M and containing no surface of discontinuity. Around the point M, let us draw a closed surface small enough so that it does not penetrate any body of the system other than the body i , and that it does not meet any surface of discontinuity. Let us denote this surface. Let 2 be the part of the body i not included in the surface 0. Let us imagine that all the electric charges, located outside the surface 0, act on the point M, charged with one unit of electricity, according to Coulomb's laws. The force exerted has as components the quantities X, Y, Z, given by the equalities (4), which can still be written as follows: - da db de, ox ' In these expressions, the sign \] which relates to the triple integral CIIAP. IV. - LEMMAS OF GAUSS. 29 applies to all bodies in the system other than body i; the sign ^ which relates to the surface integral applies to all discontinuity surfaces in the system; the triple integral not subject to the sign \^ extends to all elements in space 2. These expressions (5) can be transformed. Let us consider the sphere of radius i having the point M as center; let du be an element of the surface of this sphere. Let us take the contour of this element as the director of an infinitely loose cone having the point M as its vertex. Two spheres described from the point M as center with radii /' and {^r + dr) cut out on this cone an element of volume r-dlldr; we can adopt, for space 2, this mode of division into elements of volume. We can therefore write, instead of the equalities (5), Y = . . . , the last sign V indicating a summation that extends to all elements JS of the sphere of radius i having the point M for center. Now suppose that the surface contracts so as to vanish at point M by some series of shapes. The first two terms of the expression of X remain invariant during this change. The third term formed by an integral whose limits and elements remain finite, when the surface vanishes, tends to a finite limit whose value does not depend on the series of shapes through which the 0-surface passes. Thus X also tends to a finite limit, independent of the series of shapes through which the 0-surface passes, and this limit can be represented by the first of the equalities (4). An analogous demonstration applies to Y and Z and allows us to state the following proposition: If Von assumes that Coulomb's law always expresses r mutual action of two electrified particles, no matter how small the distance between them, Vaction 3o BOOK I. - ELECTROSTATIC FORCES. exerted on a point charged with one unit of electricity retains a definite magnitude and direction, even if the point is part of an electrified volume, provided that it is at a finite distance from any electrified surface of discontinuity. The components of this action are represented by the equalities (4). We will also see that, under these conditions, each of the three components of the force is a continuous function of the coordinates of the point on which the force acts. Let us consider two neighboring points M(^,y, z) and M'(^', _/', z'). Around the point M, let us draw a closed convex surface, 0, enclosing also the point M'(x', jk', -z'), and meeting no discontinuity surface. Let X be one of the components of the force exerted at point M. We can write X = Xj -1- A2, X, being the component of the action exerted at the point M by the electricity spread inside the surface 0, and X2 the component of the action exerted at the point M by the electricity outside the surface 0. An analogous notation allows us to write, for the point M', We will therefore have A. - A. =^ X. - Xj -H Xq - X2. The reasonings made previously show us that, for any point M" inside the surface 0, we can write .R dx x'i^SjT pâ'^'-^2, said being a surface element of the surface of radius 1 having for center the point M", and R the distance to the point M" of a point of the surface having its spherical perspective svir the element t/S. This expression shows us that we can always take the surface small enough, to have at any point M" inside this surface |x';i<^ ("), (' ) The notation a, borrowed from M. Weierstrass, means absolute value of a. CHAP. IV. - LEMMAS OF GAUSS. 3l r, being any positive quantity given in advance. We will have, in particular, |x,|<|, ix;i<5. The surface being thus chosen, let us notice that X2 is obviously a continuous function of the coordinates of the point M, so that one will be able to trace around the point M a sphere S, entirely included inside the surface 0, and small enough so that one has |X;-X,|<^; whenever the point M' is inside this sphere. We see then that, all the times that the point M' will be inside the sphere S, we have |X'--X| having, obviously, the same value for all the points equidistant from the center O of the spherical layer and the action being obviously directed along the line led from the center O to the point M. Let us agree to count positively the action when it is directed from O to M and negatively when it is directed from M to O. Let us take a spherical surface S having for center the point O and passing by the point M. Let r be the radius of this surface. It is (' ) Newton, Philosophiœ naturalis principia mathematica. Liber I, Sectiones XII, XIII. 36 BOOK I. - ELECTROSTATIC FORCES. It is easy to see that, for a similar surface, the quantity Fj, defined as we have just done, will have precisely, at any point, the value $. We will thus have (i5) CFn^S = 47r/'2<ï.. Two cases are now to be distinguished: 1° The point M is located in the cavity inside the spherical layer. In this case, all the electric charges from which the actions Fjj emanate are external to the surface S. According to equality (i4) we have § F:, ^S = o, or else, by virtue of equality (i5), 4> = G. A homogeneous spherical layer does not exert any action on the points enclosed in the cavity that it delimits. 2° The point M is outside the spherical layer. In this case, all the electric charges of this layer, which produce the action Fj,-, are inside the surface S. We have therefore, according to equality (14)5 S being the sum of these charges, and equality (i5) becomes 311 * = z r A homogeneous spherical layer acts on an external point as if all the charges it contains were gathered in its center. These two theorems easily allow us to calculate the action exerted on a point located in any way in space by a sphere on which electricity is distributed in homogeneous concentric layers. In particular, if the electrified point is outside the sphere, the sphere must act on it as if all CHAP. IV. - LEMMAS OF GAUSS. 87 the electric charges it contains were gathered in its center; the action exerted must therefore vary in inverse proportion to the square of the distance of the point from the center of the sphere. We know that Coulomb experimentally verified this consequence of his law by observing the oscillations of a small electrified body in the presence of an electrified conducting sphere. 38 BOOK I. - ELECTROSTATIC FORCES. CHAPTER Y. PROPERTIES OF THE POTENTIAL FUNCTION AT A POINT INSIDE THE ACTING CHARGES. § 1 - Existence and continuity of the potential function at a point inside the acting charges. In Chapter II, we defined the potential function and studied its properties only at a point located at a finite distance from all acting charges. We will now extend the results obtained, modifying them as necessary, so that they become applicable to a point placed in contact with the acting charges. In the present chapter, we will consider a point M placed in a region where the solid density of electricity is not zero, but located at the same time at a finite distance from all surfaces on which a surface distribution exists. Provided that the solid density has, at point M, a finite value, we know, from what was said in the previous chapter, that the action exerted by all the electric charges of the system on a charge equal to the unit supposedly placed at point M has perfectly determined compcsants X, Y, Z, which are given to us by the equalities (4) of Chapter IV Around the point M, one can draw a closed surface S, simply connected and convex, bounding a volume at all points of which the solid density of electricity exists and has a finite value. This volume does not contain any surface of discontinuity. Let us denote this volume by the index i and the rest of the space by the index 2. The electricity spread in the region i exerts at the point M an action whose components are Xj, Y,, Z, ; the electricity spread in the region 2 exerts at the point M an action whose com CHAP. V. - PROPERTIES OF THE POTENTIAL FUNCTION. 3() are X2, Yo, Z2; we have f X = Xi-+-X2, (0 Y = Y, + Y,, \ lu ^ Li\ H- ^2* The quantities Xo, Yo, Z2 obey the laws indicated in Chapter II; in particular, if we denote by x^ y, z the coordinates of the point M, and by Y^i^iy-, ^) the potential function at the point M of all the electricity spread in the region 2, we will have I A2 - - £ - - > 1 Ox z - t "*^'. L>t - - £ -^ - - \ oz Our study must be about the quantities X,, Y,, Z,. Of these quantities, we know nothing, except that they are finite, determined and represented by the equalities Xi = / / / -~r - - dai dbidci, 'JJJ, '■' "^^ ( 3 ) /Y, =JJJ-t ^ da, db, de, , p dr -2 dy p dr fim duidbidci, p being the solid electric density at a point of the element datdbtdct, taken in the region i, and r the distance from this point to the point M. Let us surround the point M with a closed surface S {Jig. 5) simplc Fig. 5. related, entirely contained in S, and which can, by a 4o BOOK I. - ELFCTROSTATIC FORCES. Let 3 be the region between the surface 2 and the surface S. Let us consider the quantity (4) "'"///' da^dh^dcz, This quantity tends to a finite and definite limit when the surface S tends to vanish at the point M passing through any series of shapes. Let us consider a ray vector emanating from point M. It first encounters the surface S an odd number of times, at points whose distances from point M are rj, rg, ..., r^p^ ''ap+i ; it then encounters the convex surface S at a single point whose distance from point M is R. Let us suppose that this ray vector is one of the generators of a cone having point M as its vertex and spherical aperture t^w. We can easily see that we can write J = ^ c/to / pr dr -+■ I pr dr -h . . . -h i pr dr -h j pr dr\. Now suppose that the surface S vanishes at point M. All the integrals in square brackets tend to o, except the last one which tends to a finite limit ^R / prdr. The quantity J thus tends towards a finite and determined limit J'= V d(x> I pr dr. If we refer to the expression (4) of J, we see that, when the surface S fades at the point M(j:,jk, ^), the quantity J tends to a finite and definite limit, which we will denote byY i {^,y, ^) ^) Aa? which, according to equality (5), can be written 44 by putting BOOK I. ELECTROSTATIC FORCES. Now we have 1 j. = [r^- 2 (a - x) ^x -I- Aa;2]2. -(A rr' {r ■+■ r) Aa? and we also have the inequalities \a - x\%r, \a - X - Aa? I ^ r', I A^ 1 1 /- + /-', the latter resulting from the fact that the absolute value of A^ is the length of the third side of a triangle whose other two sides have lengths r and r' . These inequalities, together with the previous equality, give Aa? \ r r'{r-hf^') 7-r' If /- is less than r', the second member is less than - ^^ If /- is at least equal to r', the second member is at most equal to -7^ - We have therefore, in any case Ax \ r the triple integral extending to the entire volume inside the surface. 48 BOOK I. - ELECTROSTATIC FORCES. The three equalities we have just written give ^^^^ S ^ ''^"^"- ""fff^"^"^"^^ ^"*■ This is the equality we wanted to establish. § 3 - Existence of second order partial derivatives of the potential function at a point inside the acting charges. So far we have only admitted that one could, around the point M (^, jv', z), draw a closed, simply connected and convex surface S, such that the solid density p of electricity exists and has a finite value at all points inside the surface S. In the present paragraph, we will further assume that e/i all these points the solid density of electricity is a continuous /function of £C, y, ^, admitting, with respect to these variables, partial derivatives of the first order which are Jinies. Let us always denote by i the region of the body interior to the surface S, and by 2 the exterior region {fi g- 8). Fig. The first of the equalities (i4) gives us easily ('7) dx -m^ p dr^ dx dui dh\ dc\ Let us draw, around the point M, a surface S, included in S, separating the region i into two other regions: one, 3, interior CHAP. V. - PROPERTIES OF THE POTENTIAL FUNCTION. 49 to S; the other, 4j, between S and S. We will then have I ///è £ ^'"^ ^^' "^"^ = ///^ £ ^""^ "^^^ ^'^ (i8) < First, we can always take the surface S small enough to have (19) I fjf^ ■£ ^^3 ^*3 of r, being a positive value given in advance. Secondly, it is easy to see that "S" cos(v, x) dl, 1). - I. 5o BOOK I. - ELECTROSTATIC FORCES. The quantity /// - --' - da'^ Ml dcL r dai is the potential function at point M of an electric mass which would be spread out in space 4 so as to have ■-- for density solid at each point. 11 It follows from what was said in § 1 of this Chapter that one can always take the surface S small enough to have If we denote by d^ the spherical opening of the small cone having for vertex the point M and for directrix the contour of the element ) dx dy dz, extended to the volume enclosed by this surface, which is a positive quantity, contrary to what we have just demonstrated. Therefore, if, in the domain of a point, the solid electric density exists, is finite and continuous; if, in this domain, the partial derivatives of the second order of the potential Junction exist, are finite and continuous, we have, at this point (25) AV = - 4Trp. This equation, called Poisson's equation, contains as a special case the equation AV=o, given by Laplace for the point outside the acting charges. § 5 - History. The equation (aS) was discovered in i8i3 by Poisson ('). The (') Poisson, Remarks on an equation which appears in the theory CHAP. V. - PROPERTIES OF THE POTENTIAL L.V. FUNCTION. 53 The demonstration that Poisson gave at that time applies strictly only to bodies inside which electricity is distributed in a homogeneous way. If a point ;M is situated inside such a body, we can take it as the center of a sphere which divides space into two regions: one i, inside the sphere; the other 2, outside this sphere. We then have The Laplace equation gives us AVo^o, and, on the other hand, the calculation of AV), easy to do by means of Newton's theorems on the attraction of homogeneous spherical layers, gives AV2 = - 4-":?, which demonstrates equation (aS) for a uniformly electrified volume. When the volume surrounding the point considered is not uniformly electrified, we can, according to Poisson, take the radius of the auxiliary sphere to be small enough to be able to consider the electrification inside this sphere as homogeneous and reproduce the previous demonstration. This demonstration obviously lacks rigor. Poisson realized this and later gave two other proofs of the important theorem he had discovered. The first of these two demonstrations ( ' ) is the one we have just given in the preceding paragraph; it is based on the use of Gauss's lemmas, which Poisson demonstrates like Gauss. Although Poisson does not mention Gauss by name, it is probable that he was aware of the latter's published work on the attraction of ellipsoids; for, as Mr. Bacharach (-) rightly remarks, he knew that Gauss was the author of the lemma of the ellipsoid: des attractions des sphéroïdes ( Nouveau Bulletin de la Société philoniathique de Paris, vol. III, p. 388- 892; i8i3). (') Poisson, Mémoire sur la théorie du magnétisme en mouvement {Mémoires de l'Académie des Sciences pour l'année 1828, t. VI, p. 455-463. Paris; 1827). (') Max Bacharach, Abriss einer Geschichte der Potentialtheorie, p. n {Inauguraldissertation.yfiirzho\xT%; i883). 54 BOOK I. - ELECTROSTATIC FORCES. "It is difficult to suppose that Poisson, who was so deeply concerned with the problem of the attraction of ellipsoids, as evidenced by the important works he published on this subject, was still unaware after thirteen years of Gauss's work, which had not long become famous. " The second Poisson proof ( ' ) is based on the properties of Laplace functions. Poisson admitted, without bothering to demonstrate it, the existence of the potential function and its partial derivatives of the first and second order within the acting charges. Gauss was the first to notice this gap and tried to fill it in a Memoir (-) which is of capital importance for the theory of the potential function. Gauss showed (*), by the method we have reproduced in § 1 of this Chapter, that the integral qvii defining the potential function at a point outside the acting charges continues to represent, inside the acting charges, a finite and continuous function of the coordinates. Gauss ('') also gave an analogous demonstration about the integrals which, within the acting charges, represent the first-order partial derivatives of the potential function. This demonstration was reproduced by Dirichlet (■'), Riemann (") and Heine (^). But it is not enough to show that these integrals continue to represent continuous functions of the coordinates inside the acting charges, to be able to assert that these functions (' ) Poisson, Mémoire sur l'attraction des sphéroïdes {Connaissance des Temps for 1829, p. 354-364. Paris; 1826). (^) Gauss, Allgemeine Lehrsàtze in beziehung auf die im verkehrten Verhàltnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungs- Kràfte {Resultate aus den Beobachtungen des magnetischen Vereins im Jahre, 1839. Published by Gauss and Weber in Leipzig in iSjo. - Gauss, Werke, t- V, p. 197). C) Gauss, Allgemeine Lehrsàtze, § G (Gauss, Werke, t. V, p. 202). ('-) Gauss, ibid. (5) Lejeune-Dirichlet, Vorlesungen ûber die im umgekehrten verhâltniss des Quadrats der Entfernung wirkenden Kràfte, published by F. Grube, § 4. Leipzig; 1876. (0 Riemann, Schwere, Elektricitât und Magnetismus, edited by K. Haltendorfi, § 6. Haltendorfi", § 6. Hanover; 1876. (' ) Heine, Handbuch der Kugelfunktionen, t. H, § 17. CHAP. V. - PROPERTIES OF THE POTENTIAL FUNCTION. 55 continuous are still, inside the acting charges, the three derivatives of the potential function. Clausius (' ) was the first to give a proof of this last proposition; but his proof relies on the use of an integration by parts, the very legitimacy of which presupposes the correctness of the proposition to be proved. The first satisfactory demonstration of the theorem in question is due to Bouquet and can be found in Briot's Théorie mécanique de la chaleur (-). The demonstration we have given is due to M. Otto Hôlder ("). The existence of second derivatives of the potential function was demonstrated by Gauss (^) in much the same way as we have indicated in this Chapter. This demonstration, like the one given by Clausius {^) essentially assumes the existence of partial derivatives of the electric density in the neighborhood of the point considered. Mr. Otto Holder (®) has shown that the partial derivatives of the potential function still exist in the much more general case where only the following assumption is made: Around the point M.[x^y^z), we can draw a domain such that, for any point M'(.r',y', z') of this domain, we have lA . \9{x,y,z')-p{x,y,z)\{x",y',z")-?{x',y\z')\''-<'-s^^-=° The quantity and its partial derivatives of the first order being continuous in all space, we obviously have so that equalities (3) and (4), added member to member, give the integral extending to the whole space. 11 It is easy to see that this equality (5) could not be satisfied if we did not have, at all points in space, the equality -(.")' ^m-m-' that is - 16) from from ^=°' toi = '' Let us imagine, in fact, that at a point M in space the quantity F^ is not equal to o. Since this quantity F^ is a continuous function of the coordinates, one could always, around the point M, delimit a domain A small enough, so that the quantity F^ is different from o at any point of the domain A. Let B be the space outside the domain A. We would have Ç f Ç ¥'^dxdydz= f f Ç Y'^ dx dy dz -\- C Ç Ç Y^dxdydz. In the second member, the second integral could only be zero or positive; the first would certainly be positive; the integral in the first member could therefore not be equal to o as required by equality (5). Thus, the function is a continuous function in all space, admitting, in all space, partial derivatives of the first 62 MVRE I. - ELECTROSTATIC FORCES. order which, according to the equalities (6), are identically zero. The function therefore has the same constant value in all space; since, moreover, the quantity %{x, y, z) must tend to o when the point [x, y, z) moves away from the surface S indefinitely, the quantity must be identically zero, and the two functions V and V must have the same value at all points in space. This is the very proposition that we stated. The properties that we have listed as belonging to the potential function of an electrified volume are therefore particular to this function and cannot belong to any other function. We can give them the name of criteria of the function in question. It is sufficient to be sure that a function has all of them (') to be able to affirm that it represents the potential function of an electric distribution inside the surface S and adjusting at each point the solid density This important theorem is due to Lejeune-Dirichlet (-), who used it in the study of the problem of the attraction exerted by a homogeneous ellipsoid. § 2 - Potential function of a homogeneous ellipsoid. The study of the action exerted by a homogeneous ellipsoid on a point outside or inside this ellipsoid has solicited the efforts of almost all the great geometers who have succeeded one another since Newton until our days.) Newton, Maclaurin, d'Alembert, (' ) Instead of stating the criteria of the potential function relative to infinity, we will often be content to say that the potential function cancels at Tinfini, whenever no confusion is to be feared. (' ) Lejeune-Dirichlet, On a general means of verifying the expression of the potential relative to any mass, homogeneous or heterogeneous {C relie' s Journal. Bd. XXXII, p. 80-84; 1846). (2) The history of the problem of the attraction of ellipsoids can be found very completely in C. Paraira: Over de Methoden ter bepaling van de aantrekking eener ellipsoïde op en willekeurlig punt {Academisch prœfschrift. Amsterdam; 1879). A more summarized history of the same problem serves as an introduction to Chasle's Mémoire? Mémoire sur l'attraction des ellipsoïdes ; solution synthétique pour le cas général d'un ellipsoïde hétérogène et d'un point extérieur. Paris: 1846. CHAP. VI. - CRITERIONBI.V OF THE POTENTIAL FUNCTION. 63 Lagrange, have initiated the solution of this problem. It was given by Legendre and by Laplace. Ivorj, Gauss, Poisson, Chasles and many others have solved it by different methods. The solution of Lejeune-Dirichlet (' ) differs from those whose authors we have just enumerated, in that it could not have served as a method of invention. Lejeune-Dirichlet takes, in fact, the function which, according to the work of his predecessors, should represent the potential function of a homogeneous ellipsoid, and, by using the crileria established in the preceding paragraph, he ensures that this function is indeed the potential function of the ellipsoid considered. This method is obviously less satisfying for the mind than the beautiful and elegant methods of invention proposed by various geometers, and in particular by Gauss and by Chasles. But it has an advantage over all the others which makes us prefer it, brevity. Either 372 v2 32 (7) - -^-r? + - - 1 = ^ a^ 0^ C the equation of the ellipsoid related to its center and axes. Let's consider the equation a?2 y2 ^2 The quantity u^ which appears in this equation (8), is, according to this equation, an algebraic function of .r, y^ z. It is easy to see that this equation, reduced to the integer form, is of the third degree in z/. It thus gives, in general, for w, three determinations. Suppose we have the inequalities " > 6 > c. The three determinations of u separate from each other the (' ) Lejeune-Dirichlet, On a general means of verifying the expression of the potential relative to any mass, homogeneous or heterogeneous {Crelle's Journal. Bd. XXXII, p. 8-84; 1846). - Vorlesungen, etc., edited by F. Grube; 2" Part. Leipzig; 1876. 64 BOOK I. - ELECTROSTATIC FORCES. four quantities - a% -b^, - c2, -4-00. The substitution theorem leads immediately to this result; indeed For M = - 00, the first member of equation (8) has the sign - "a = Or - Y)" "M = - a^ -t- 7) " -+ " a = - 62 - -^ " j) - " ii = - b^-\- Ti " V -+ W M =: - C2 V) " 1^ - "M = - c'^ -\- ■r\" " -I )) M = -i- ce " )) Consider the largest of the three roots, the one between - c- and +3o; this root is positive if the point [x,y,z) is outside the ellipsoid, zero if the point (^,y,z) is on the surface of the ellipsoid, and negative if the point {x,y,z) is inside the ellipsoid. To demonstrate this, it is sufficient to note that, by the substitution i^ = o, the first member of equation (8) becomes x^ y- 2^ positive, zero or negative quantity, depending on whether the point [x^ y., z) is outside the ellipsoid, located on the surface of the ellipsoid or inside the ellipsoid. The sum of the three roots of equation (8) has the value I [(372-1- JK'--i- ^2) _ ("2 + ^2+ C2)]. This sum grows beyond any limit when the point (^, y, z) moves indefinitely, in any direction, away from the center of the ellipsoid. Therefore, the root between -c^ and -h 00 grows beyond any limit when the point {x, y, z) moves indefinitely away from the ellipsoid. With these preliminaries in mind, let us consider the expression y'i , . -.r, V 7 f v^'-f-X è2_^_X C2-1-X (9) N{x,y,z)=^T.ç>abc f tT ^^' CHAt>. VI. - CRITERIA OF THE POTENTIAL FUNCTION. 65 in which U has the value o if the point (jc,y,z) is not outside the ellipsoid, whereas, if this point is outside the ellipsoid, U is the positive root of equation (8). If the point (x,y,z) is located on the surface of the ellipsoid, these two rules lead both to take for U the value o. The quantity p is the uniform electric density inside the ellipsoid. Finally the radical is taken in absolute value. The integral in this expression (9) has an infinite limit; but this integral is not illusory, because, for infinite values of )v, the quantity under the sign / is of the order of laughter We will show that the function \(x^ y, z) has all the criteria of the potential function of the ellipsoid, and that, therefore, it represents this potential function. 1" The lower limit of integration, U, represents two different analytic functions of .r, y^ z, depending on whether the point [x^y^z) is external or internal to the ellipsoid^ but these two analytic functions take the same value o at a point on the surface of the ellipsoid; U is therefore, throughout space, a uniform, finite and continuous function of x^ r, z. Moreover, the element under the sign / is also, in all space, a uniform, finite and continuous function of x^ y, z\ because none of the quantities a2-t-X, 62^X, c2^À becomes equal to o between the limits of the integration. 11 It follows that, in all space, the function V(.r, jk, z) is a uniform, finite and continuous function of x, y, z. 2" Let R be the distance of the point [x, y, z) from the origin of the coordinates; let us show that, when R increases beyond any limit, the product RV remains finite. We have Rr.__:^_ J^ îL 1 , f I a^-r-X b^-^l c2-f-Xj j. v/(a2-^X)(62-f-X)(c2-4-X) r Tzp abc I If we refer to the definition of U for an external pointD. - I. 5 66 BOOK I. - LUS ELECTROSTATIC FORCES. the ellipsoid, we can see that we can pose the following v being a function of x, y, z which remains finite, positive, and does not tend to o when R grows beyond any limit. Let us then make a change of variable defined by the equality 1 = R2(p+ t), and we will have ^1 R2 "2 62 c'^ 1 , R .+^-^ Ri+ ^-^^ ^^ml^ RV=Trpac>c / - - dt, v/(ë- )(l^'- )(l-^--) When R grows beyond any limit, ^> -^j .^ do not grow beyond any limit; v is always finite, positive, and does not tend to o; ^:i tt-? t^t tend to o; VR therefore remains finite. ^ K^ K"' K'^ 3° Let's calculate the derivative -r-ax The formula of the difierenliation under the sign / , applied to the equality ('9), gives = TTP abc X àx ' \j "M-X v/(a2_HXy(62_HX)(c2+X) ' ^ a2_t_ X "~ b^'^^ ~ c2 _^ X d\J \ v/(a2+X)(62 + X)(c2-hX) Jx = t^)' Two cases are to be distinguished: If the point [x^y^z) is inside the ellipsoid, U has the value constant o, and so is -r-; the second term of the factor in braces is equal to o. If, on the other hand, the point {x,y^ z) is on the surface of the ellipsoid or outside the ellipsoid, we have . x^ y^ z^ _ ^^^^ "^ 6Mnj"^ ^M^ - I - o, CHAP. YI. - CRITERLV OF THE POTENTIAL FUNCTION. 67 el second term of the factor between braces is still equal to o. The previous formula is thus reduced, in all circumstances, to dx - ~paoc I X /(a^+X)(62+X)(c2-+-X - > T- have analogous expressions. R" -r: remain finite when R grows beyond any limit. 4" The formula of differentiation under the sign / , applied to equality (10), gives us / d'^\ , r ^ + =- 1 dl 1 -T - = - ■ntpabc I - - - yJ-^- J a2_|_X v/(a2+X)(62H-X)(c2+X) US dS, dS, cr - - - db, fi Let us take a point M(œ,y,z), which can be made to tend in a continuous way towards one of the surfaces S of the system. Let us suppose that, from this point M, we can lower on the surface S a normal M pi whose length tends towards o and whose orientation varies in a continuous way when the point M tends towards a point of the surface S; this will always be possible if, as we shall suppose, at the point P of the surface S, towards which the point M tends, and at all the points close enough to the point P, the surface S admits a tangent plane, and if the orientation of this tangent plane varies in a continuous way from one point of the surface S to another. Let [j. {Jig'. lo) be the foot of the normal in question. We can Fig. 10. always assume that the point M is so close to the surface S that the pomt [JL is as close as possible to the point P towards which the point M tends. CIIAP. VII. - ELECTRIFIED SURFACES. 78 Through the point M let us lead a plane T parallel to the tangent plane of the surface S in ij.. If the point jjl is sufficiently close to the point P, we will always be able to draw a closed line L on the surface S, cutting out on this surface a limited area Si which contains inside it the point P and all the points [x relative to all the successive positions of the point M and which, moreover, has the following property: A normal A a to the plane T will always meet the area S, at a point at most a and will always make a finite angle with the tangent plane at a. This being said, we will observe that the force exerted at point M by all the electrified surfaces of the system always results from two forces: one generated by the electricity distributed on the cap S, ; the other distributed on the rest of the electrified surfaces. The latter obviously remains a uniform, finite and continuous function of the coordinates of the point M, even if the point M tends towards the point P of the area S, or comes to cross the area S, ; let us not occupy ourselves with it any longer and study the former. The first has a component directed along the normal N to the surface S(, normal counted in a determined direction, the direction piM. This component, which we will call Fj, will require our attention. By jjlM, let us lead two planes normal in ^ to the surface S4, making between them an angle d. Maiï at z Rz R2 dz (R2-4-^2)2 (R2_i_32)2 (R2-i-^2)2 '^^ We have therefore, by applying to the expression of Fj^ the formula of integration by parts and by designating by S, 3b, JL^ what become 0-, z, V in a point of the curve L, .27: ^Si. ^in ^i)\. - ^ r ^ "1 , ,. Note that this transformation of the integral giving Fjj essentially assumes that the point M is not located on the CHAP. VII. - ELECTRIFIED SURFACES. 75 surface S; otherwise the formula of differen dation on which it is based would lose all meaning. But the second member of the formula (2) has, as we shall see, a finite value, which varies in a continuous way when the point M approaches the surface S( on a given side of this surface and even when it comes to rest on the surface Sf. However, this value, which represents F^ as long as the point M is not on the surface S), however close it may be to this surface S, can cease to represent F^ at the moment when it comes to rest on the surface S, . What we have just stated is obvious for the first of the three terms that appear in the second member of equality (2). Consider the second of these terms. 11 can be written J. J. ■-■-■-[,.(£)■]"""' Under the assumptions made, all the factors of which the quantity under the sign / is composed remain finite in the field of integration, even if the point M comes to lie on the surface S|, except the term R -. The theory of integrals in which the quantity under the sign / becomes infinite then shows us that the integral keeps a finite value and varies in a continuous way when the point M approaches the surface Sj or comes to be placed on this surface. An analogous demonstration applies to the last term of the second member of equality (2). So we arrive at the following consequence: Let us consider a surface S in a region S| of which the tangent plane varies in orientation in a continuous way while the curvature has a finite value [this condition is equivalent to the existence of the derivative of cos(N, v), as can be seen easily]. Suppose further that at any point in this region Sj the surface electric density has a finite value, varies 76 BOOK I. - ELECTROSTATORY FORCES. in a continuous way and admits partial derivatives with respect to the parameters which fix the position of a point on the surface S. Let us take a point M situated in the vicinity of this surface on a given side of this surface. The action exerted by the electrified surface on this point M admits a component following the normal to the surface which will join the point M. When the point M tends in any way towards a point P of the surface, this normal component F^ tends towards a finite and well determined limit. This limit varies continuously with the position of the point P on the surface. It is not allowed to say that this limit represents the value of F^ on the surface. Let us take two points M, M' located on either side of the surface S, and infinitely close to the same point P. In this case, the direction iN' is approximately opposite to N; the direction v' to v. The quantity cos(N', v') has therefore approximately the same value as the quantity cos(N, v). Similarly z and z' have approximately equal values and opposite signs. It seems therefore that, when these two points tend simultaneously to the point P, the two quantities F^ and F^ must tend to equal limits and of opposite sign. But, if we observe that the quantity which appears under the sign / in the last two terms of the second member of equality (2) becomes infinite in the field of integration, it is easy to see that this conclusion is no longer necessary. Thus, when two points, situated on either side of the surface Si, both tend towards the same point of this surface, the normal components of the forces exerted on these two points tend towards limits which are not necessarily equal and of opposite sign. Let us take the point M on the surface Si ^fig. ii)* around the point M, on this surface S), let us draw a closed line / separating the area Sj into two others; a bounded area S2 surrounding the point M and an annular area S3 included between / and L. Through the point M, let us lead the normal N to the surface S,. The electricity distributed over the area S3 exerts at point M an action CHAP. VII. - ELECTRIFIED SURFACES. 77 whose component following N is T K ona I n ivi 1 /-" /-" .Rcos(aM,N ) J, Jo /-^cos(N,v) p being the radius vector of the projection of the curve / on the Fig. II. plane T tangent at M to the surface S). Now we have cos(aM, N) = -, So we have .S ./o J. (R^ '-^ dKd^. p (R2-i- 32)2 COS(N, V) As the surface S| has a finite curvature at the point M, we can write z = KR2, K tending to a finite limit when R tends to o. Then we have *N - - f f ^ dK d'^. J^ Jp (i-i-K2R2)2cos(N,v) In this form, we see that, if the curve / contracts so as to vanish at the point M through any series of shapes, which makes all the quantities p tend to o, the quantity ^^^ tends to a finite and determined limit, represented ^8 BOOK I. - ELECTROSTATIC FORCES. by the symbol r'^ cos(N, v) OR ^ -%os(r, N)^S. S. This limit is, by definition, the normal component of V action that the electrified surface S| exerts at one of its points M according to Coulomb's laws. The demonstration of the existence of this limit only assumes that the quantity j,^. 2° If the surface density is continuous, in the vicinity of the point M, the quantity <ï>j<^ varies continuously with the position of this point on the surface S. CHVP. Vir. - ELECTRIFIED SURFACES. 79 3" Let us now suppose that the surface density admits, in the vicinity of the point M, a finite derivative with respect to the arc of any curve traced on the surface S. Let M" be a point outside the surface S which tends towards the point M while remaining on the side of the surface S marked by the normal N). Let N". be a normal to the surface S going to meet this point. The component along N'j of the action exerted at point M" by the electricity spread over the surface S tends to a finite and definite limit when point M" tends toward point M in any way. This limit Fjj^ varies continuously with the position of point M. Let M' be a point outside the surface S which tends towards the point M while remaining on the side of the surface S marked by the normal No. Let N^ be the normal to the surface S going towards this point. The component along ]N', of the action exerted at point M' by the electricity spread over the surface S tends towards a finite and determined limit when point M' tends towards point M in any way. This limit Fy^ varies continuously with the position of point M. 4" The quantities F;y_, Fy^ are not necessarily equal and of opposite sign; the quantity F^^ is not necessarily equal to <ï>,<^; the quantity F^^ is not necessarily equal to - ^y^ or ^^. We will find later the value of the quantities Fn. -^ Fn,, *N. -Fn" *N, -F>v § 2 - Study of the tangential components of the action exerted at a point by an electrified surface. Let us take in the plane T {^g- i3), led by the point M parallel to the tangent plane at {x to the surface S, a given direction M0. Let us agree to count the angles "L from this direction. The action exerted at point M by the electricity spread on the surface S, admits along M0 a component F0, which has for value ^^n ^A Rcos(aM,e) ,_ , 8o LIVKE I. -- ELECTROSTATIC FORCES. But we have ces (aM, 6) = sin(aM, N) cos4', R sin (xM, N) = - , r -2 = R2-f-^2. So we have -0 -^0 (R2^_ 22)2 COS^,i>, /; But we have, by the way, cos 71 being any positive quantity. However, the first member of this inequality does not depend on 6, CHAP. VU. - ELEGANT SURFACES. 87 inequality can only take place if we have (6) C (*s.-4-Fx,-2Tr£cr)rf2 = o, the area AB being any area drawn on the surface S and containing the point M. This equality, in turn, cannot take place unless we have, at point M, ^s, -+- F.v. - aTiea = o. Suppose, indeed, that at point M, the quantity is different from o. The value of this quantity varies in a continuous manner with the position of the point M on the surface S; for, according to the hypotheses made and the consequences deduced from them, each of the three quantities ;^p Fj^^ and | be the three forces whose normal components are ¥ji^, F^,, ^ . These three forces, we know, must have the same tangential components. They are therefore located in the same plane normal to the point M on the surface S. If A2M represents the force F2, Ma the force ^ and MA the force F, {^g". 16), the two lines Ma, MA, must have, on the plane tangent at M to the surface S, the same projection MB,, equal and directly opposite to the projection MBo of the length MA^ on the same plane. i.ELECTROSTATIC FORCES. If '= R' cosO + ï' sinO; putting for R' and T' their values, it comes yy'= R cos8 + T sinO - 4'^JKCOs9, and consequently p' = p - ^T^y cos9. " ... If it is a fluid layer spread over a spheroid of any shape, and arranged in such a way that it does not exert any action on the interior points, which is the case of the electric fluid, we will have T'= o, R'= o; therefore also T = o, R = ^Tzy, from which it follows: i° that the tangential force is null at the exterior surface, as we have already proved in the preceding issue; 2° that the force normal to this surface is proportional to the thickness of the layer at each point. "This demonstration is the one we announced at the beginning of this Memoir, and which was communicated to us by M. Laplace. We have made it a little more general, by considering first a fluid or solid layer, which was not subject to exert any action on the points of its interior surface. " Laplace and Poisson considered, as we can see, the properties of a very thin layer, but not of a rigorously superficial layer. Green considered, in 1828, this kind of layer (2) and (' ) Poisson, loc. cit. p. 33. (-) George Green, An essay of the application 0/ mathematical analysis to the theories of electricity and magnetism. Nottingham; 1828 (^Mathematical papers of the late George Green, p. 3o. London; 187 1). D. - I. <7 g8 BOOK I. - ELECTROSTATIC FORCES. demonstrated, for each of its points, the fundamental relationship it ^ __ , Its demonstration, based on the use of the identity that bears its name, would require, in order to be made rigorous, that we first prove the various propositions that we have exposed in paragraphs 1 and 2. It is to Gauss (' ) that we owe the method by which it is possible to establish all these propositions in a rigorous and elegant way. Recently, Mr. Otto Holder(-) has shown that all these propositions remain true in an extended case where the electric density does not admit a derivative along the arc of the surface it covers. Finally Lejeune-Dirichlet (^) was the first to insist on the role of criteria played by the propositions that Gauss had proved. (') Gauss, Allgemeine Lehrsdtze, n°' 12, 13, 14, 15, 16 (G.vuss Werhe, Bd. V, p. 212). (") Otto Holder, Beilràge zur Potentialtlieorie. II" Abschnitl {Inaugural Dissertation. Stuttgard; 1882). (') Lejeune-Dirichlet, Vorlesungen iibcr die im umgekehrten Verhàltniss des Quadrats der Entfernung wirkenden Kràfte, p. 65. Leipzig; 1876. CIIAP. VHI. - nVIM'EL bK QlKLyi KS PlUNCII'liS I>K MKCVMQli:. Qi) CHAPTER YIII. REMINDER OF SOME MECHANICAL PRINCIPLES. § 1 - Statement of the principle of virtual velocities. In this Chapter, we shall briefly recall some of the notions of Mechanics which we shall have to deal with frequently in the rest of these Lessons. We shall begin by tracing the statement of the principle of virtual velocities, which, as Lagrange has shown, dominates the whole of Statics. Let us consider a material system and, to fix ideas, let us suppose that it is formed by a finite number of material points separated from each other by finite distances. Let M, (xi, j,, ^,), Ms (^2,^25 -S2), - - -, M"(.r",jK", Zn) be these points. This system is subject to certain bonds; these bonds are of two kinds. Some are expressed by an equality or several equalities between the coordinates of one or several points of the system. This is, for example, for a point, the condition of remaining on a surface or on a curve. We will call them bilateral links. The others are not susceptible of being expressed by one or more equalities. Let us suppose, for example, that the point M, (.T), j^t, z-i) is constrained to remain either outside a body, or on its surface, without being able to penetrate its interior. Let the equation of the surface of this body; suppose that inside this body we have f{x,y, z) o. lOO BOOK I. - ELECTROSTATIC FORCES. The bond imposed on the Mt point is then expressed as follows: /(^i, Ji, 5i)|o. Such a link, in what follows, will be called a unilateral link. Suppose that between the points of the system there are p bilateral connections /l(^l, Jl, -1, -^2, ...) = 0, / . 1 Ai^i,ri, -1, a-2, ...) = o, and q unilateral connections (2) ?2(^1, JKl, -Zl, ^2, --- ) = 0, ?l(j^UyU -21, -2^2, ...)=0. Suppose that the quantities ^i, 7u -Si, ^2, JK2, ^îi - - ) - - " - - " verify conditions (i) and (2); suppose we choose the infinitely small quantities 8a7i, oji, 0^1, 8a72, Sj2, 2^2, . . . , - - - ) - - - ) S^", Sj/j, S^/i, such that the quantities a^ï+oj^a, J2+0J21 -32-1-3^2, Xn + Sa^", 7/, -H S/", ^" + 8^rt CHAP. HIV. - UVPPEL DK SOME PRINCIPLES OF MECHANICS. lOI also satisfy conditions (i) and (2): ùxi, 07,, 0^1, 03^2, 0/2, 032, constitute the components of the motion of the various points in a virtual displacement of the system; or, more briefly, constitute a virtual displacement of the system. A virtual displacement o.r,, SjKi, . - will be reversible if - ôJCj, - oj'i, . . . is also a virtual displacement of the system. If all the bonds to which the system is subject are bilateral bonds, all the virtual displacements of the system are reversible displacements. Indeed, in the case where the system is subject only to bilateral links, for 8x,, ôj-,, S2<; 3^2, . . . to constitute a virtual displacement, it is necessary and sufficient that these quantities verify the relations àfx . ^/i . àfx . toZn àfp ^ àf" ^ atfp . 4- j- àZa = 0, and these relations are obviously also verified by the system of quantities - 0J7,, - oji, -ozi; -0X.2, Let Xi, Y,, Z, be the components of the explicitly given force acting on the point M, ; X2, Y2, Z12 be the components of the explicitly given force acting on the point M2; .... By definition, in any virtual displacement ô^Tj, ôj/"(, . . . ), the first force performs a virtual work Xi Ixx -\- Yi 071 -+- Zi Szi ; the second force does virtual work X2 tX^ -4- Y2 072 -H Z2 0^2 î etc, .... The Principle of Virtual Work, often also called 102 IJVRK I. - LKS ELECTROSTATIC FORCES. The principle of virtual velocities can then be stated as follows: For a system to be in equilibrium, it is necessary and sufficient that, in any virtual displacement of the system, the sum of the virtual works of the given forces applied to its various points is zero or negative. In other words, the necessary and sufficient conditions for the equilibrium of a material system are obtained by writing that, for all systems of values of ô:^;^, oj^,, os,,, ... that constitute a virtual displacement, we must have (3) Xi QXi -+-Yi Ô/i H- Zj 0^1 -f- \2 0^2-1- - - . -H Z,j OZii 1 o. If all the virtual displacements of which the system is susceptible are reversible, this principle is expressed simply by the equality (4) Xi oa^-i-f-Yi o/i -I- Zi03i-H X2 0X2+ . . . -t- Z" oz,i= o. It is in this last form that the principle of virtual work was first stated by Jacques Bernoulli ('); it is also in this form that Lagrange, in ^ Analytical Mechanics, made it the foundation of the whole Static. The more complete form expressed by the inequality (3) is due to Gauss (^); it has been exposed with great care by Clausius (^) and by M. Cari Neumann (^'). Except for Sturm's Treatise, no French Treaty of Mechanics mentions this complete form of the principle of virtual works. (' ) Letter quoted in Varignon, Nouvelle Mécanique. {') Gauss, Ueber ein neues allgemeines Grundgesetz der Mechanik {Crelle's Journal, t. 4; 1829. Gauss, Werke, Bd. V, p. 27, in note). - Principia generalia theorice figurée fluidoruni in statu œquilibrii (Nouveaux Mémoires de Goettingue, vol. VII; j83o. Gauss, Werke, Bd. V, p. 35). - Letter from Gauss to Moebius quoted by M. Cari Neumann. (' ) Clausius, On the potential function and the potential, trans. Folie; Paris, 1870. (") C. Neumann, Ueber das Princip der virtuellen oder facultativen Verriickungen. CHVP. Vin. - REMINDER OF SOME PRINCIPLES OF MECHANICS. Io3 § 2 - Statement of d'Alembert's principle. Fundamental formula of Dynamics. The principle of virtual velocities, which is the foundation of all Statics, can also be used to establish all the theorems of Dynamics, with a modification that is deduced from the Principle of crAlembert. Let us consider a system formed by n moving points, M, Mo, . . . , M". At a certain time t, the coordinates of these points are ^i, yi, ^\, for Ml, ^2, 72, -2, for Ma, ^n-, JKrt, -", for M". This system is subject to certain connections. These connections can be bilateral, as it happens if a point is subjected to move on the surface of a certain body, or unilateral, as it happens if a point is subjected only to not penetrate inside a certain body. We will leave aside the case of unilateral connections, which is studied in the theory of percussions, and we will suppose that the system is exclusively subject to bilateral connections, expressible at each instant by equalities between the coordinates of the various points of the system at that instant. Two cases can arise for each of these equalities: either the form of the relation which links the coordinates of the various points of the system and expresses a bilateral link remains the same at all times; this is what happens, for example, if a point is constrained to remain on a surface which remains fixed during the motion of the system; or the form of this equality varies from one moment to the next: this is what happens, for example, if wine point is constrained to remain on a surface which moves at the same time as the system. In the first case, the link considered can be expressed, for the whole duration of the movement, by a relation between the coordinates of the various points of the system I04 BOOK I- - ELECTROSTATIC FORCES. In the second case, the link considered can be expressed, for the whole duration of the movement, by a relation between the coordinates of the various points of the system and the time: The linking equations can therefore explicitly contain time. Consider a system of n material points M, , M2, . . . , Mrt whose coordinates at time t are a:) , jVi 5 s, ; ^2? - - , ^wThis system is subject to p bilateral links expressed by equations that can depend explicitly on time [ /l(""l,JKl, Zi,X2, ..., t) = o, *' I ■ l fp(^i,yu ^1,^2, . . ., o = o Let's give t, in these equations, a certain constant value to, and let's put Fl(^\,yU -1, 372, ... , Zn) = /iCa^i, JKl, ^1, T.2, ..., Zn, to), F2(^l,ri>-lJ^2,-- .,Zn) = Ai^U 7l,^U ^2,- - -,2,1, to), (6) F/^OljJKl, -1,3^2, --■,Zn) -fp{Xi,yi,Zi,X2, ..., S", ^o). Let m^ , ma, . . . , mn the masses of the n points M, , M, . . . , M^ ; let X) , Y, , Z, the components of the force explicitly given which, at time tç, acts on point M, ; let ^^ , ^^, ^ the components, at the same instant, of the acceleration of this point; let us adopt analogous notations for the points M^, ..., M", and we will be able to state D'Alembert's principle in the following way: Whatever the instant to that Von considers in the duration of the motion, if Von were to take a system formed by n points M, M2, ..., M", placed without initial velocity in the positions they occupy at V instant to', if Von were to subject the point M, to a force whose components would be ^^-'"'^rf^' ^'-'"--^' ^"-'"'-J^^ CH.VP. VlII. - REMINDER OF SOME PRINCIPLES OF MECHANICS. lOf) the Mo point to a force whose components would be ^'--""'■^i^' ^^-'"^^' ^'--'"'--d^' etc., ...; if finally one subjected the system thus constituted to the links F'i(^i)^i) ^1) x^, .... z,i) = o, (7) '^ the i^iiyij ^1) ■^2) - - - ) -") - o, /e system thus formed would be in equilibrium. This is the principle of d'Alembert who thus reduces any problem of Dynamics to a problem of Statics. This principle leads, with the help of the principle of virtual works, to a formula from which we can deduce the equation of any problem of Dynamics. Here is how we can obtain this formula. The necessary and sufficient conditions for the equilibrium of the fictitious system we have just considered are expressed by the principle of virtual velocities; if therefore ôX|, ôy, , 05|; ùX2, -- - designate vm virtual motions compatible with the links ('j) which are all bilateral, we must have d-z,. dt^ This formula (8) summarizes, as we have said, the equation of any problem of Dynamics. It is called the fundamental formula of Dynamics. It is due to Lagrange, who made it the foundation of the whole study of Dynamics in his Analytical Mechanics. § 3 - Various remarks on the connections. Equality (8) must take place for all virtual displacements compatible with the bindings (7); in other words, it must take place for any system of values of 0x1, Sj^j, 3s,, . . ., that lo6 I.IVRIÎ 1. - ELECTROSTATIC FORCES. checks the equalities ()F, . c)Fi , dF, , f)Fi , àxt 4ri àzi dx2 dF^. ^F, . dF^. dF^^ (9) \ Oa-i àfv ^-Si dx^ 1 ) f OFp . dFp , (^F/. > , (^F/> > v (^37, Oyi -^ dzi dx2 The (/> H- 1) linear and homogeneous equations (8) and (9) must be verified by any system of values of oxt, oj^,, oz,, . . . that verify equations (9). Equation (8) must therefore be a consequence of equations (9). The theory of linear equations teaches us that it is necessary and sufficient to find p factors 1,, \o, ..., "Xp, independent of Sjc,, ùyt, oz, ; 8^2? - -1 such that by multiplying by X, the first member of the first equality (9), by ^.o the first member of the second... . by "kp the first member of the last one, and adding the results obtained to the first member of the equality (8), we obtain an identically zero sum. Therefore, there must be/? factors depending only on .r, , j'i , ^i ; jco, - - -, ^n, t, such that we have, whatever O^i, Zy^, ùZi ; ^x^, - - -, oz,i, the equality v" r /v -^ ^F, . dFj . dF" d-'xq\ . ()Fi . ÙF. . ÙF" d'-yq\. '/ = ! dy,, dy,, dyn df^ /" , JFi . ÔF^ . dFp d'-z,,\^ -] which requires (that we have the 3/i equalities , t)Fi . c)F2 , cJF,, d-'zi Zi-+- A, -- -h A2 T- -^.--+Xp- -^ = /"i- ,:r> CHAP. VlII. - REMINDER OF SOME PRINCIPLES OF MECHANICS. I07 These equalities show us that the point M, for example, moves as if, without being subject to any linkage, it were subjected not only to the explicitly given force whose components areXi, Y|, Z,, but also to a force whose components would be A, -^ + Aj , dF, ■-^^''' dz, This fictitious force is called Aç. binding force applied to point M|. The functions \^, Xo, ...,)v^ of the (3 /i -|- i) variables ^r,, >^i, ^1, x-i^ ..., Zni t are called Lagrange multipliers. Let us make one more fundamental remark about the connections to which a system can be subjected. Any virtual displacement 0;ri, Sj^o ^^"j - --? '^^n of the system is subject to the bonds (6). It must therefore verify the equalities (9), which become, by virtue of the definition of the functions F,, Fo, . -, F^ given by the equalities (6), -f- OJ?i -H -f^ OKi 4- -/- O.Si -I- . . -tàXy, C*yi ^^1 (10) { dxi dfi -' Ozi Of, ^ ttZn = 0, at <)U Oi" = o> OZa df" toz,. o^" = 0, àJP^ 4/1 ^-^1 t having in/, , f-^. . . . , /"p the value ^q; we obtain all the virtual displacements at time -f- ? - - - ? -y- , t. -^ ^ dt dt dt dt The system is subject to bonds that are not explicitly time-dependent /i (-^1, Xi, -Si, - - -, -") = o, .h C-^i, yn ^1 z,i) = o, fpi^uji, -i> ---> -") These yo relations allow us to express the 3n coordinates .r,, ji, ^,, ...^z-,i in terms of k = Sn - p suitably chosen independent variables a,, a2, ..., y./,. By replacing the coordinates by their expressions in terms of these variables, we will transform the quantity 7 = " \ ( X,/ da^q -+- Yq dyq -\- Z^ dz,, ) in Pi rfa, -f- Pa d%. ^- . . P/, d^,,. and equation (i 3) will become (i4) P,^/a,-i-P2^a,+ ... + PA../:^/,= ^( 2 "//''''- Each of the quantities P,, Po, . . , Pa is a function of the variables a,, a2, . . , a/i BOOK I. - ELECTROSTATIC FORCES. The quantity Pj doii-h P^doiz-T- . . -r- Vi^doL/c is not, in general, the total differential of a uniform function of the independent variables a,, a^, . . ., a^. But it so happens that this exceptional condition is realized in most applications; in other words, in most cases of interest to the physicist, there exists a uniform function of a,, ao, ..., ayt U(ai, a2, ..., a/,) such that we have Pifi^ai-i- Parfa,-!- . . . -t- P^-rfa/,. = - r7U(ai, "2) - - -, ^k) In this case, equation (i4) simply becomes ^U(ai, "2, ..., ■Xk) If we denote by a,, a-x-, ---, "* the values of a,, a,, ..., a^ at the time T, and by a',, a'^, . . ., a\ the values of the same variables at the time T', by adding member by member the equations, analogous to the previous one, which we can write for all the times dt included between T and T\ we find q=n q=n 7=1 7=1 Thus, in this case, the equation of the living forces is integrated when we know the initial and final positions and the initial and final velocities of the various points of the system, without it being necessary to know the sequence of states through which the system has passed between the two instants that we consider. The quantity ,/ = l is a homogeneous combination of the second degree of da\ da\ da). 'dT^ ~dt ' ■ ■ ' ~df ' CUAP. VIII. - REMINDER OF SOME PRINCIPLES OF MECHANICS. It3 Equalion (i5) is thus a first order differential equation between the k variables a, , a2, . a/t, whose values at each instant determine the position of the system. The equation of the forces x'ives thus provides, in this case, a first integral of the equations of motion of the system. The function U(a,, ao, ..., ^k) is what we will call the potential of forces X|, Y,, Z,, .... This name is due to Gauss. Lagrange, who first insisted on the importance of this function, in his Analytical Mechanics, did not give it this particular name. Hamilton and Jacobi gave the function ( - U) the name of force function; W. Thomson and Rankine named the function U V potential energy and Clausius Vergiel. If the quantity % ( Xr/ dx,, 4- Y^ dfq -+- ZqdZc, ) 7=1 is, whatever dxq, dyq, dzq, the total differential of a uniform function of the quantities Xq^ yq^ Zq, the quantity Pi dxi -{- Po d(x.2 + ...-+- P/cdxi will be a fortiori the total differential of a uniform function of a,, ao, ..., a/(. We then say that the system admits of itself a potential. But it can happen that the quantity Pi dxi -+- Po rfao -+- . . -i- Pa <^a/,. be the total differential of a uniform function of y.^^ :>.<,,... ., aA, without the quantity q = l is the total differential of a uniform function of the quantities .rq,yq^ Zq left independent. It is then said that the system admits a potential by virtue of the bonds imposed on it. In the case where the acting forces admit a potential either by themselves or by virtue of the bonds imposed on the system, the D. - I. 8 Il4 BOOK I. - ELECTROSTATIC FORCES. principle of virtual velocities can be put in a new form. In this case, we have V {X^ox,j->r Y, 07,/ -H Z^oz^) = Pioai + P2 3a2+ . + P/fôa/,-, '/ = > 5a,, ôa2, ..., ocL/f not being linked by any relation, and, moreover, by designating by U the potential P _ ^U p _ <^U P, - ^u so that the principle of virtual speeds can be stated as follows: For a system whose acting forces admit a potential to be in equilibrium in a given state, it is necessary and sufficient that, in this state, we have --- oai -H -r - oag + . . . + --- oa/t- = o, r>ai oo(.=i "a/,. whatever 8a,, oao, ..., oa^; in other words, it is necessary and sufficient that we have We can see that, in order to search for the equilibrium states of a system which admits a potential, we have to write the same equalities as if we wanted to search for the maxima and minima of the potential. § 6 - Criterion of the stability of the equilibrium. This remark is completed by the following theorem: If, for a given state of the system, the potential is minimum, this state is a stable equilibrium state. Let's start by defining exactly what is meant by this last word. We can always assume that the parameters a,, ao, . . ., T-k are equal to O for the state of the system that we consider; because, if they were equal respectively to .A.,, al)o, . . ., A>kt it would be enough CHAP. VIII. - REMINDER OF SOME PRINCIPLES OF MECHANICS. Il5 to take as new parameters to define the state of the system the parameters for the condition to be verified. Let us suppose that a system is in equilibrium for. 21 = 0, a2 = o, ..., a/^. = 0. Suppose, moreover, that the quantities Xf, y^, ^1, . -, 5" are expressed as continuous functions of a,, a^, ..., a^. We will say that the equilibrium position of the system is stable if it fulfills the following conditions: Let A,, Ao, ..., A/t" f^ be any positive quantities. We can always find positive quantities a,, a^, ..., ";t, . . . , U(j, . . . ) small enough to have |a,(01^Ai, {t)\o, except for the particular case "1=0, a2=o, ..., a^.= o. This being said, I say that we can give to the quantities "i, a>, ..., ak, ..-, iiq^ ... values small enough so that none of the valuables a,, aa, ..., o(yt can, at any time t^ reach its limit A,, Ao, ..., A;;^. Let us suppose, in fact, that at a certain moment f, one or more of the variables a,, a2, ..., a^ reach their limit. At this moment, U will have a positive value which cannot be lower than a certain limit; we can write q - n ^ U[a,(^"), ",(io), ...,"/.(fo)]-4- ^"Hull!^ '/ = i <-/ = " = U[a.(0, "2(0, ..., "a(0]+2^~' The second member will be a positive quantity that cannot be less than some limit. On the contrary, one may choose a,, "2, " -■, <^ki - - M Uq^ . . . small enough that the first member is smaller than some positive quantity given in advance, in which case the preceding equality will become absurd. It is thus clear that we can impose on the quantities a,, "25 ' - 1 cik^ - - -■) Uq^ . . . values so small that none of the quantities aj, a2, . . ., a;t can at any time reach its limit; as we have announced, equilibrium is stable. This important proposition is due to Lagraiige (' ); the beautiful and rigorous demonstration that we have just read is by LejeuneDirichlet (2), (' ) Lagrange, Mécanique analytique, P" Partie, Section III. ( = ) Lejeune-Diriciilet, Ueber die Stabilitàt des Gleichgewichts {C relie' s Journal, l. XXXII, p. 85; 1846). CHAP. IX. - OF THE ELECTROSTATIC POTENTIAL. CHAPTER IX. OF THE ELECTROSTATIC POTENTIAL. The electrostatic forces determined by Coulomb's law will provide us with a remarkable example of forces that admit a potential. Let us consider two material masses, of very small dimensions, m and m' , respectively carrying electric charges q and q' . Let /- be the distance from a point M of mass m to a point M of mass m' . Let x, y, z be the coordinates of point M, and x'^ y, ^' the coordinates of point M'. The point M is subjected to a force coming from the mass m' , whose components are ,x - x' and the point M' is subjected to a force from the mass m, whose components are ,x'-x Y' = ^^^"^' Let us suppose that the point M experiences a displacement Bx, 8y, 85, and the point M' a displacement ùx', oy', oz'; the sum of the works of the two forces that we have just considered will have the value Xox -h Y oy -^ Zoz -h X'ox'-+- Y'ùy'-h Z'oz' jl8 BOOK I. - ELECTROSTATIC FORCES. OR, according to the relationship X.OX -hYùv -hZoz -^X'^x'-i- Y'8v'-i- Z'os'= t ~{- or. From this equality, we immediately deduce the following consequence: when a certain number of electrified points are moved in relation to each other, each keeping an invariable electric charge, the work done by the forces acting between these various points, according to Coulomb's law, is equal, to the nearest sign, to the variation of the quantity The sign 2, represents the sum of all combinations, such as ^^ that can be made in the system, each of these combinations being taken once and only once. The quantity W thus represents, according to the definitions given in the preceding chapter, the potential of the electric forces; we shall give this quantity the name d.ç. electrostatic potential; it is often given, in the Treatises, the name electric energy, which we shall not use. This quantity is susceptible to a remarkable expression. Suppose that we multiply the charge q successively by all the quantities - supplied by the other electrified points of the system, and that we add all the products obtained, so as to form the expression Tt in which the sign \ indicates a summation that extends to all the charges of the system other than the charge q] then, that we repeat the same operation on all the charges of the system and that we add together all the results obtained, in order to form the expression CIIAP. IX. -- OF THE ELECTROSTATIC POTENTIAL. II9 the first of the two signs N^ extending to all charges of the system; it is obvious that, in a similar addition, we will have each of the distinct expressions listed twice Qn' > -^ ^ and that, therefore, we will have the sign 2, having in the second member the same meaning as in equality (1). But, on the other hand, if V is the value of the potential function at the point M where the charge q is located, we have (3) Ilf-V. From equalities (1), (2), (3) results this new fundamental equality (4) ^=V^qN. Up to now, in order to study the electrostatic potential, we have assumed that electricity is concentrated in isolated points, but we can just as easily assume that it is spread over volumes or surfaces. In this case, if the various parts of the system undergo displacements during which all the elements of the electrified volumes and all the elements of the electrified surfaces preserve invariable electric charges, the work carried out by the electrostatic forces is, to the nearest sign, the imriation of a potential W, the expression of which can be obtained in the following way: For each of the electrified surfaces, we form the integral J QVaJS, (7 being the surface density at a point of the element â?S and V the value of the potential function at the same point. For each of the electrified volumes, we form the integral l j 1 j ^?dxdydz, 120 BOOK I. - ELECTROSTATIC FORCES. p being the solid electric density at a point of the element dxdy dz and V the potential function at the same point. Finally, all the integrals obtained are added together. Thus, we have (5) ^=\^^'^'d^+^Jjj^9dxdyd^, This expression is susceptible to very important transformations that we will indicate. We know that we have and Equality (5) can therefore be written This equality (6) shows that W is determined when we know the value of the potential function V at all points in V space. Green's theorem will allow us to give a new form to equality (6). In order not to complicate the calculations without any serious advantage for the generality of the demonstrations, we will assume that the system contains a single electrified body A {fig. 19) and a single surface Fig. 19. the surface S, which separates this body from the non-electrifiable medium which surrounds it. The directions N,, No become then res CIIAP. IX. - ELECTROSTATIC POTENTIAL. MI pectively the directions N/, Ne, of the normal to the surface S directed towards the interior or exterior of the body A. Under these conditions, equality (6) is written W = - C f Ç\ \y dxdydz Inside the body A, let us draw a surface S' infinitely close to the surface S. Let A' be the space enclosed by the surface S'. Let N' be the normal to the surface S' towards the interior of the space A'. Green's theorem [Chapter III, equality (5)] gives us fjfy AV dx dy dz + g V ^^^, dS' Let us now make the surface S' tend to the surface S; the es dY pace A' tends to the space occupied by the body A; x^ tends to -T=7- and the previous equality becomes (7) Outside the body A let us draw a closed surface S" enveloping the body A and infinitely close to the surface S. Let N" be the normal to the surface S" to the unlimited space surrounding it. Let B" be this unlimited space. By a reasoning analogous to the one which gave us equality (4) in Chapter VI, we find Let us now make the surface S" tend to the surface S; the space B" tends to the unlimited space B outside the surface S; -rzp BOOK I. - LKS ELECTROSTATIC FORCES. tends to -r^ ? and the previous equality becomes Equalities (6a), (y) and (8) give the following formula the integration that appears in the second member extending to the whole space. This is the important expression of the electrostatic potential that we wanted to achieve. This expression shows us, first of all, that the electrostatic potential can never be negative. Can it be equal to o? We have seen, in § 1 of Chapter VT, that the equality led to, in any point of the space, the equality V = o. From then on, at any point inside one of the bodies that form the system, we have and at any point of the discontinuity surfaces that bound these bodies, we have __ i_ /^ toS_ Therefore, for the electrostatic potential to be equal to o, it is necessary and sufficient that the whole system is in the neutral state. The electrostatic potential of any electrified system is positive. Second, if we assume that each volume element of space is subjected to forces whose potential would be CIIAP. IX. - nor ELECTROSTATIC POTENTIAL. 123 All these forces would have the same potential as the forces which, according to Coulomb's laws, act between the various electrified particles of the system; these forces would thus perform, in any virtual displacement of the system, the same work as the forces given by Coulomb's laws. It follows that we could substitute these forces for the forces given by Coulomb's laws. What are these forces, whose potential on each volume element is known to us alone? This is a question which gave rise to important work by Maxwell (' ), but which we shall not examine in the present Volume. (' ) Maxwell, Traité d'Électricité et de Magnétisme, V" Part, Chap. V. See also H. Poincaré, Électricité et Optique : I. Les théories de Maxwell et la théorie électromagnétique de la lumière, Chap. II. BOOK II. L\ ELECTRICAL DISTRIBUTION ON CONDUCTING BODIES AND THE LEJEUNE-DIRICHLET PROBLEM. CHAPTER ONE. CONDITION OF ELECTRICAL EQUILIBRIUM. - ELECTRICITY RESIDES ON THE SURFACE OF CONDUCTING BODIES. § 1 - Principles of Poisson's theory. So far we have left invariable the electric charge taken by each of the elements of a system; we have assumed that each of the elements could move, but that the state of electrification of none of them could vary. When a body is supposed to satisfy a similar restriction, it is said to be a bad conductor. If, at a given moment, a poorly conducting body has no electricity at any of its points, it will have none at any moment. The non-electrifiable medium in which the bodies we will study are supposed to be immersed is, by hypothesis, a bad conductor. It is not a question here of knowing whether the bodies that nature presents to us and to which we give the name of bad conductors exclusively offer properties that can be represented by the preceding definition. In truth, the bad conductor or insulating body that we have just defined is an ideal body, whose properties we shall nevertheless study, even if it means comparing these properties with those of real bad conductors by experiment. In general, on bodies placed in determined positions, the electric distribution will cease to be variable only when certain conditions are verified by the electric density at each point. The bodies are then said to be good conductors 126 BOOK II. - DISTRIBUTION AND DIRICIILET PROBLEM. their, and the conditions in question are called electrical equilibrium conditions. These are the conditions that need to be set. Poisson (M sought to determine them by hypothetical considerations. We shall see, in Book III, that these hypothetical considerations do not provide a satisfactory representation of all the laws of the distribution of electricity on conducting bodies; we shall then be led to replace Poisson's theory by another theory. But the results of Poisson's theory will be preserved in the new theory. Poisson's fundamental hypothesis consists in looking at the actions studied by Dufay and Coulomb not as forces applied to the material particles that carry electric charges, but as forces applied to the electric charges themselves. This hypothesis is already contained in the very statement that Coulomb gave of the laws he discovered. He speaks there of the attractions and repulsions which are exerted between electrified bodies, or between electric charges, without making any distinction between the meanings of these expressions (-). This hypothesis is not without serious logical difficulties. A force being, by definition, the product of a certain acceleration by a material mass to which it is applied, to speak of forces applied to electric charges is to admit, a priori, that electric charges are of a certain (') Poisson, Mémoire sur la distribution de l'Électricité à la surface des corps conducteurs, read on May g and August 3, 1812 at the Académie des Sciences (. S'avan fs étrangers, 181 1, p. i). (") Coulomb, Second Mémoire sur l'Électricité et le Magnétisme, in which it is determined according to which laws the magnetic fluid, as well as the electric fluid, act, either by repulsion or by attraction {Mémoires de l'Académie des Sciences, 178,5, p. 579). At the end of this Memoir, Coulomb expresses himself as follows: "From the foregoing researches it will result: 1° That the action, either repulsive or attractive, of two electrified globes and, consequently, of two electric molecules, is in compound reason of the densities of the electric fluid of the two molecules, electrified, and inverse of the square of the distances 4° Q"<^ '^ attractive and repulsive force of the magnetic fluid is exactly, as in the electric fluid, in compound reason of the direct of densities, and inverse of the square of distances of the magnetic molecules. " {Loc- cit., p. 611.) CH.VP. I. - CONDITION OF ELECTRICAL EQUILIBRIUM. 12/ quantities of a material fluid. This is, moreover, the hypothesis generally adopted at the time of Coulomb and Poisson. If Ton does not want to make this hypothesis, if one wants to consider the electric charge as a simple parameter of which one alTects any electrified material molecule, one can no longer attribute any meaning to the mol force applied to an electric charge. We shall later find a way to free ourselves from this hypothesis which equates electric charges with material masses; for the present, we shall accept it without discussion and <'n deduce the consequences. From this assumption Poisson deduces the condition of electrical equilibrium on a system of conductors, a condition that is stated as follows: For V electricity to be in equilibrium on a conductor or on a system of conductors, it is necessary and sufficient that V electricity spread on these bodies exerts neither attraction nor repulsion on any point taken at random in the interior of any of these bodies, this point being supposed to have an electric charge equal to V unit. It is assumed that electric fluids can be used as points of application for the forces given by Coulomb's laws, and that they are free to move inside each conducting body. Therefore, for an electric charge placed at a point inside a conducting body to remain immobile, it is necessary and sufficient that it is not subjected to any force. From this, Poisson's principle necessarily follows. Let us consider, in fact, a point taken inside a conducting body; either there is in this point a certain quantity of electricity, or there is no free electricity in this point. If there is a certain electric charge at this point, for this charge to remain there and for the distribution to be permanent, it is necessary and sufficient that the action exerted on this charge and, consequently, on a charge equal to the unit placed at the same point, by all the positive or negative charges of the system, is equal to o. If there is no free electricity at the point considered, we can always 128 BOOK II. - DISTRIBUTION AND THE PROBLEM OF MANAGEMENT. days conceive that there are at this point two equal electric charges of opposite signs pi and - p.. If the charges distributed on the system exert on the quantity i of electricity supposedly placed in this point a certain action F, directed in a certain way, the first of our charges will undergo an action ^F in the direction of the force F; the second will undergo an equal action and of opposite direction. If the action F is null, both charges will remain at the point considered, which will remain in a neutral state; but, if the action F is different from o, the positive charge at the point considered will be pulled in one direction, the negative charge in the opposite direction, so that the electric distribution on the body considered will be modified. This is how Poisson's hypotheses lead from Coulomb's laws to the principle of electrical equilibrium on conducting bodies. The uncertainty of the hypotheses in question leads to an equal uncertainty for the principle of electrical equilibrium, which does not possess the certainty of Coulomb's laws. We will now study the analytical consequences of this principle and compare them with experience. This will be the object of this and the next book. § 2 - In the state of equilibrium, electricity resides on the surface of bodies drivers. Coulomb was the first to state (' ) this proposition: "In a conductive body, charged with electricity, the electric fluid spreads over the surface of the body, but does not penetrate into V interior of the body. " This proposition. Coulomb gave it as a consequence of the experiment; but, in reality, none of Coulomb's experiments directly demonstrates the proposition in question, and (' ) Coulomb, Quatrième Mémoire sur l'Électricité, in which two principal properties of the electric fluid are demonstrated: the first, that this fluid does not spread in any body by chemical affinity or by elective attraction, but that it is divided between different bodies brought into contact solely by its repulsive action; the second, that in conducting bodies the fluid, once it has reached the state of stability, is spread over the surface of the body and does not penetrate into the interior {Mémoires de l'Académie pour 17S6, p. 72). CHAP. I. - CONDITION OF ELECTRICAL EQUILIBRIUM. 129 No experiment can demonstrate it directly, because any point inside the volume that a conductor occupies is inaccessible. The preceding proposition can only be established theoretically. The possibility of theoretically establishing this proposition has already been recognized by Coulomb. "This property of the electric fluid, he says (' ), to spread over the surface of conducting bodies and not to penetrate into the interior of these bodies when this fluid has reached a state of equilibrium, is a consequence of the law of repulsion of its elements, in inverse proportion to the square of the distances, a law which we found in our first Memoir ; but, ('Oinme it was experience, and not theory, that led us, we thought we should follow the same course in the exposition of our researches; let us now see how the theory generalizes the result announced by experience. Let us now see how the theory generalizes the result announced by the experiment." This is followed by a demonstration, inaccurate moreover, of which we will have to speak again later. In spite of this indication by Coulomb, Poisson does not seem to have bothered to deduce from his theory the demonstration of this fact that electricity in equilibrium resides only on the surface of conducting bodies; he always takes this truth as a consequence of experience (-). Green was the first (^) to show that this truth was a simple and immediate consequence of Poisson's theory. Green's demonstration can be replaced by the following: Let us suppose, for a moment, that inside a conducting body there is electricity distributed either on certain surfaces with a finite surface density, or in certain volumes with a finite solid density. Let's draw inside this conductor a closed surface S, subject to meet the surfaces supposed to be electrified only along certain lines. To this surface, let us apply the general consequence of the lemmas of (' ) Coulomb, Loc. cit, p. 76. (' ) Poisson, Mémoire sur la distribution de l'électricité à la surface des corps conducteurs, p. 2 (Memoirs of Foreign Scholars, 181 1). - Second Mémoire sur la distribution de l'électricité à la surface des corps conducteurs, p. I (Ibld.). (' ) Green, Essay... (Nottingham, 1828. Mathematlcal papers 0/ the late George Green, p. 22). D. - I. 9 l3o BOOK II. - DISTRIBUTION AND ROBLIiMK OF DIRICIILET. Gauss, established in Book V, Chapter Vil, § 3. JNoiis will have S ¥s,dS = /iT.tD]l, where D is the total electrical mass enclosed within the surface S. But, according to Poisson's principle, if the electrical equilibrium is established on the considered conductor, the electricity which resides in the whole system exerts a null action at any point taken at random inside this conductor. We have therefore, at any point of the surface S, B\. ::= o, and, consequently, the previous equality becomes DR - o. Thus, within a conductor, there is no electricity when equilibrium is established; electricity resides exclusively at the surface of the conductor and the insulating medium. Coulomb {*) thought that the property we have just demonstrated could still be exact, even if the actions that are exerted between two electrified points were not ])as inverse to the square of their distance. It was sufficient, according to him, for this property to remain exact, that the distance /- tending towards o, the electrical action grew less rapidly than - - This thought of Coulomb's does not conform to the truth. If we admit the principle of electrical equilibrium proposed by Poisson; if we admit, on the other hand, that the electricity in equilibrium on a conductor is carried exclusively on the surface of this body, we can deduce that the mutual action of two electrified particles is in inverse proportion to the square of the distance separating them. Indeed, from the hypotheses made, and from an obvious reason of symmetry, it results that an electrified conducting sphere must be covered with an electric layer having at any point the same density, and that this layer exerts a null action at any point inside the sphere. But can a homogeneous spherical layer, in another law than the law of nature, exert a null action at any point which (' ) Coulomb, Loc. cit, p. 75. CIUP. I. - CONDITION OF ELECTRICAL BALANCE. l3l li is interior? Laplace (') showed that it could not, and M. J. Bertrand (-) gave an ingenious demonstration of this impossibility which we shall reproduce. Let the magnitude of the repulsive force between two electric particles located at the distance /-. If the product i''f{i') remains constant, the proposition is proved. Let us suppose then that this product varies and goes, for example, increasing, when /- increases from po to 0|. Consider a sphere of diameter (in + P() {/ig- 20) and rccou ( 20. ^ ^ 5 / ^ M > ^\/ y pX/ ^_ Let's say that the sphere is filled with electricity, which forms on its surface a homogeneous layer of density u. Let us express that the action of this electricity at a point inside the sphere, for example at point M, whose distances to the various points of the sphere vary between po and pi, is equal to zero. Through point M, let us lead a plane AB, normal to the diameter CD which passes through this point. It divides the sphere into two caps, and the action of one, ACB, must balance the action of the other, ADB. The action of the cap ACB is obviously directed along the diameter CD. This action is the sum of the compo (') Laplace, Mécanique céleste, Part I, Book II, n° 12 ( i"^" edition, t. I, p. i4o.) (') J. Bertrand, Loi des actions électriques {Journal de Physique pure et appliquée, 1" série, t. II, p. 4i8; 1878). M. Liouville was the first to demonstrate that the law of the inverse square of the distance is the only one in which electricity is carried to the surface of bodies. See J. Bertrand, Note sur quelques points de la théorie de l'électricité (Liouville's journal, vol. IV, p. 49^! iSSg). l32 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. The action of each of the elements PQ of the spherical cap ACB on the point M is measured along this diameter. Now the component along MO of the action of the element PQ is, by designating by dtii the angle under which from the point M we see PQ and by N the normal to this element towards the outside of the sphere, cos(N, r)'' Let us consider the component along MG of the action exerted at the same point M by the element P'Q' of the cap ADB, this element P'Q' being cut by the same spherical opening cone c/o). It has the value May so we have '!>'= sîT - r /('- )cos(r', MD). cos(N', r j-' ^ ' ' ' cos(/-, MG) :--- cos(/-', MD), cos(N,/-) = cos(N', /-'); *' _ r\f{ r' ) ^ /- and /■' are between po and o^ and ;-' is greater than /'. Thus we have *' > *. The actions of the element P'Q' and the element PQ have a positive component directed from M to G. All the elements of the sphere can thus combine two by two in such a way as to give a positive component directed from M to G. The point M is thus subjected to a force directed from M to G. As a result, we are led to a contradiction if we do not suppose that we have r^/{r) = const. CHAP. U. - N AND A SINGLE STATE OF ELECTRICAL BALANCE. l33 CHAPTER I THERE IS ONE AND ONLY ONE STATE OF ELECTRICAL EQUILIBRIUM. § 1 - Analytical translation of the Poisson principle. Let M be a point inside a conductor on which the electrical equilibrium is established. This point is external to the acting masses, since the electricity resides exclusively at the surface of the conductor. The first derivatives of the potential function exist at this point and are related to the components of the force by the relations X - - - ^^ Y - - - '^^ Z - - ' ~ dx Oy ùz But, according to the equilibrium condition given by the Poisson theory, we have X = o, Y = o, Z = o. We have therefore, at any point inside a conductor, d\ d\ d\ - ■ - - o . - o, . Ox at dz Thus, when electrical equilibrium is established on a system of electrified conductors, the potential function (i a constant value at V inside each of the conductors; conversely, if this condition is met for each of the conductors, V electrical equilibrium is established. Two comments on this principle. In the first place, the potential function, being constant at infinitely close points on the surface of the conductor and being continuous in all space, has the same value at all points on the surface of the same conductor. Secondly, if N/, N^ are the two directions, interior and exterior, of the normal at a point on the surface of the conductor, the surface density o- is given at this point by the relation l3i BOOK II. - DISTRIBUTION AND PROBLEM OF DIRICIILET. general àY âV Now, in the present case, the first order partial derivatives of the potential function are zero at any point inside the conductor, even if it is infinitely close to the surface. We have therefore and, therefore, à\ ^'^ OX.^-'"' § 2 - Is there a state of electrical equilibrium? With these two remarks in mind, here is the question we will first consider: A system consists of a number of bodies placed in given positions; some are poor conductors /'t have a given electrical distribution; others are conductors and each of them contains a given total electrical charge. Can this charge be distributed on the surface of each of them in such a way that electrical equilibrium is established on the system? To answer this (juestion, we will first demonstrate the following proposition: An electrical distribution that makes the electrostatic potential maximum or minimum^ is an equilibrium distribution. Let us suppose, indeed, that for a certain distribution, the electrostatic potential w ^^ tS ^^ be maximum or minimum; suppose that the distribution undergoes an infinitesimally small variation 5 the variation undergone by W must be equal to zero. II. II. - IN KT A SINGLE ELECTRICAL BALANCE KTAT. l35 However, a variation of the distribution must not alter either the distribution on each bad conductor body, or the total charge each conductive body; any variation in the distribution is thus obtained by superimposing a number of operations analogous to the following: An electric charge dq passes from a certain point M of a conductor to another point M' of the same conductor. Therefore, the variation of W must be zero for any such operation. Suppose, first, that the charge at point M decreases by dq^ all other charges remaining constant. W would obviously increase by o,\s^-^zdrj^'-L=-z\dq, V elanl hi value of the potential function at point M. Then suppose that, with things in this state, the charge at point M' increases by dq^ all other charges remaining constant. W would obviously vary from V ('is the initial value of the potential function at point M', and (V' + dV') what this value becomes when the charge at point M decreases by dq. Now we have obviously MM Or has therefore ÔW = 3i W -r- 5,W =z(\'-Y)dq-t -EL- . MM' By neglecting the second order smallest t ^ and by MM' equals oW to O, we find V'=V, which shows (jue the potential function has the same value at any two points inside the same conductor and, consequently, that any electrical distribution that makes W )3G BOOK II. - DISTRIBUTION AND DIRICIILET PROBLEM. maximum or minimum is, as we said, an equilibrium distribution. Let us now consider the system for which we want to prove that there is an equilibrium distribution. Whatever the electrical distribution on this system, we know that W can be expressed as follows [Liv. I, Ch. IV, equality (9)], the triple integral being extended to the whole space. This expression shows that the quantity W is never negative. Whatever the electrical distribution on the conductors that form the system, it cannot become less than a certain limit; the quantity W, considered as a function of the N' Let W be the potential function of the charges distributed on the bad conductor; let U be the potential function of the charges distributed on the good conductor in the first distribution; let U' be the potential function of the charges distributed on the good conductor in the second distribution. We have \ v-=u+\v, ( 'I ) ', I V'= U'-+-W. Conducted at any point of the body good conductor we have jn;- + jn; = "' the ('galities(3) become Cî) _L [ÊR . ^iL I /âU' àl]'\ 47r [d'yi,'^ O^i)' With this in mind, let us consider the equality \(a'-a)(lJ'_U)rfS, Green's theorem allows, as we have already seen in Book T, Chap. IX, to replace the second member by r.///! r-^-T- [^-T- [^-Ti "---- integration extending to the whole space. On the other hand, by virtue of the equalities (4), we have. in all space, U'- U== V- V, ICVP. II. - LN AND A SINGLE ELECTRICAL EQUILIBRIUM STATE. I ÎÇ) it follows that, on the surface S, (U' - U) takes the constant value (V, - V,). The first member of equality (6) can thus be written (V',-V,)g(a'-cr)rfS,, which is equal to o, by virtue of either equality (î), or equality (2 bis). The equality ((3) becomes which supposes that we have in all the space OR _ dv' di] _d{y dv _ diy à.r O.r Oj- Oy Oz and, therefore, by virtue of the equalities (5), T = a' at any point on the surface S,. This equality proves the stated theorem. By another method, Gauss ( ' ) had already proved this theorem for a single conductor removed from any influence. The general demonstration that we have just read is due to Liouville (-). (') Gauss, Allgemeine Lehrsâtze... {Gauss Werke, Bd V). (') LiouviLLE, Note à l'occasion du Mémoire de M. Chasies, Addition à lu Connaissance des Temps pour i84j. l4o BOOK II. - DISTRIBUTION AND PROBLEM OF DIRICIILET. CHAPTER m. THE GAUSS IDENTITY AND THE ARITHMETIC MEAN THEOREM. § 1 - The Gaussian identity. In this and the following chapters, we will quickly review some of the most beautiful analytical ideas that geometers have used to solve the problem of electrical distribution. Gauss's identity (') has been used as a starting point for important theories. Let V be the potential function of a system of electric charges M', M", M'", . . . concentrated in points F, P", P'", Let, on the other hand, v be the potential function of electric charges m'j m", m'", . . . concentrated in points />', p" , p'", .... At the points P', P", P'", . . ., the function v has the values v', v", ç'", At the points/"', p", //", .... the function V has the functions V, V", ' 5 - - - - We have obviously _ _M^ M^ M'" I" M' M" M" Y" = - - -I- -f- - P'"" P // P and of even, v' m' -1 m" + /n'" - p'I p "V' v" m' -t m" -1 m" />'"P" p'P" Let's consider the q uantity M'<.''-f -M' 'v'-^-W" v'" -f- . (' ) Gauss, Allgemeine Lehrsâtze . , art. 19 {Gauss Werke, Bd V, p. 221). cii.vp. m. - the identitiî: of galss, etc. i4i Its value is - M />'!" p "V' p"'V' m' m" m" I m Vp'~p'" p"?" /)'"p" which can still be written ,/ ÎVr M' M" \ . "Hi>' ^ p'^p^'-^---) ../ M' M" M'" \ \ ï V ^ p ï /> or /«'V'-^/«"V"-4-/n"'V"^-.... Or has the identity ( I ) M' v' -T- M" v" -t- W v'" -h . .- m' \' -+- Dr!' V" -i- m" V" -^ . . . or, abbreviated, ( I bis) ^ '^^ '' ~ ^ '" ^ - The electric charges that form one of the two systems, or that form both of them, instead of being concentrated in isolated points, can be spread over surfaces or volumes. The previous identity remains exact, with the only condition that we replace the summations by the corresponding integrations. Let us suppose, for example, that the first system is formed by a surface S, on which the electric density has a value S, and the second by a surface 5, on which the electric density has a value t. The Gaussian identity will be written §.., It is important to note that this equality is still correct, if we assume that the surface S coincides with the surface s, (jue V is the potential function of a layer of density S distributed on this surface, and v the potential function of a layer of density a distributed on the same surface. BOOK II. - DISTRIBUTION AND PROBLEM OF DIRICULET. § 2 The arithmetic mean theorem. From this identity, Gaiiss (' ) deduced a fundamental theorem to which M. Cari Neumann gave the name of Theorem of the arithmetic mean . The first system that we will consider, in order to demonstrate this theorem, is formed by any masses distributed in any way. The potential function V of this system takes a value Vo at the point i^o The second system is formed by a layer of surface density equal to the unit distributed on the surface of a sphere of rajon R having for center Pq. The potential function v of this sphere has the constant value '-^ - at any point inside the sphere or its surface; at an external point, whose distance from the point the" 11 1 1 4-R^ . is /% it has the value - r The quantity ^wV becomes here xV^/S, the summation being en (hie at the surface of the sphere. If Mo is the total electric charge which, in the first system, is inside the sphere, and if M', M", . . . are the charges which are outside the sphere or on its surface, we have M' W 1 iAU=4TrRMo^-47:R2, ,Vffô = 4-RMo+4-RM -r -^^ So we have If the sphere does not contain any electric charge of the first system, we will have Mo=o, and the previous equality becomes (' ) Gauss, Allgemeine Lehrsàtze . , art. 20 {Gaitss Werke., Bd V, p. 222). cH.vp. m. - dk gvuss identity, etc. 143 Thus, when a sphere contains no acting charge inside, the value of the potential function at the belly of this sphere is the aritlimetrical average of the values that the potential function takes on the surface of this sphere. This is the so-called arithmetic mean. § 3 Theorems on the variation of the potential function outside the acting charges. The very simple theorem we have just proved is of rare fruitfulness. Let us now indicate some of its most important consequences. 1° If the potential function has a constant value in a certain region of a linearly connected space containing no acting mass, it is constant in all this space (' ). Let be the linearly connected space contained inside the surface T (fig.-m). In this space, the surface S limits another l-ii;. ... linearly connected space a, inside which the potential function has a constant value a. Can it happen that, as soon as the surface S is crossed, the potential function takes a variable value from one point to another? If this assumption is correct, as the potential function (') Gauss, Allgemeine Lehrsàtze. .., art. 21 {Gauss Werke, ^Bd V, p. 224). i44 LIVHK H. - distribution and dirichlet problem. varies continuously from point to point, one can always find a linearly connected domain [i, contiguous to the space a, at all points of which the potential function will have either a value less than a, or a value greater than a. Let us assume, to fix ideas, that the first assumption is realized. Let us draw a sphere having its center at a point M in space a, a part ADB of its surface inside space a, and another part ACB of its surface inside space ^. The integral vVâfS, extended to the surface of this sphere, will necessarily have a value less than /\Tzl\'-a, while 4''^^'^ should be its value, according to the arithmetic mean theorem. The hypothesis made is therefore inadmissible, and the stated theorem is proved. 2" Ln a point located at a distance jinie of any acting niasse, the value of the potential function can be neither a maximum nor a minimum. Let us suppose, in fact, that the value V " of the potential function at the point Mo is a maximum value or a minimum value; a maximvim value, to fix the ideas. Around the point Mo, we can always draw a domain containing no acting force, and such that at any point M of this domain the potential function has a value V, less than Vo- If from the point Mo as center, we draw a sphere of radius R contained in this domain, the integral v V âfS extended to this sphere will necessarily be less than .^"J^R^Vo, contrary to the theorem of the arithmetic mean. The impossibility of the hypothesis made leads to the correctness of the stated theorem. This theorem leads in turn to new consequences, already stated by Gauss (' ) in particular cases and demonstrated in a completely general way by M. Cari Neumann (-). (' ) Gauss, Allgemeine Lehrsàtze ... , art. 2G, 27, 28 {Gauss Werke, Bd V, p. 228). (*) Carl Neumann, Revision einiger allgemeinen Sàtze aus der Theorie des logarithmiscJien Potentiales {Malhematisclie Annalen, vol. III, pp. 34o-344 and 430-434; 1870). - Untersuchungen ûber das logarilhmische und Newton'sche Potential, pp. 39-49. Leipzig; 1877. CHAP. m. - the identity of gauss, etc. 145 Let us consider a closed surface S inside which there is no acting charge; it can happen that the potential function has the same value at any point inside this surface, or that its value varies from one point to another in the space enclosed by this surface; in the latter case, the values it takes in the interior of this space and on the surface which limits it are all between a lower limit A and an upper limit B; as the potential function is a continuous function, there exists, in the interior of this space or on the surface which limits it, at least one point where it takes the value A and one point where it takes the value B. Such a point, where the potential function reaches its lower or upper limit, cannot be located inside the surface S, because at this point the potential function would have a minimum or a maximum. It is therefore necessarily located on the surface S. Thus, if Von traces a closed surface containing no acting mass, either the potential function is constant at any point inside this surface, or, at V inside this surface, it is constantly between the largest and smallest of the values it takes on the surface. This proposal leads immediately to this consequence: If the potential function has the same value at all points of a closed surface which contains no acting charge, it also has the same value at all points inside this surface. We will see later on the crucial importance of this proposal in the study of electrical equilibrium. Let us consider a closed surface S containing all the acting masses; the unlimited space outside this surface S can be considered as a closed space between the surface S and a surface whose points are all at infinity. If we notice that at any point of the latter the potential function has the value o and if we reason as in the previous case, we arrive at the following result: D. - I. ,0 l46 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. Let's draw a closed surface enclosing all the acting charges inside. If the potential function has the value o at all points on this surface, it has the value o in ^unlimited space cjui outside it. If it does not have the same value o at all points of this surface, let us denote by B and A the largest and the smallest of the values that it takes there, these two values being able to be equal between them. In the unlimited space outside the considered surface, the potential function is always between the smallest and the largest of the three values o, A, B. A note on the first of these two cases: From what we have said for this case, we see that, if the potential function V has the value o at all points of a closed surface containing all the acting masses, we have, at any point of this surface d\ Now, for any closed surface, we have S rfs=:^iz,m, ,')1L being the sum of the charges that the superlattice contains. We arrive at the following result: For the potential function to have the value o at all points of a closed surface which contains all the acting charges, the sum of these cliarges must be equal to o. What happens in the case where the sum of the acting masses is equal to o without having an identically zero potential function at any external point? Let T {fig. 22) be a surface which contains all the masses acting either inside it or on its surface. Let us draw a sphere of radius R containing the surface T. As we have seen in the demonstration of the arithmetic mean theorem, we CHAP. III. - the identity of GVUSS, ETC, IJ^J will have C V ^/S = 47ïRMo+ 4-R"^ ( ^' -i- ^ +. . .) . In the first member, the integration extends to the entire surface of the sphere. In the second member, Mq is the sum of the masses integrated l<'ig. 22. In the present case, we have the following values for the surface T, M', M", ... the external jasses. In the present case, we have Mo=o, M'=o, ^1"= o, cl, therefore, C V ^S =: o. According to this equality, if the function V is not equal to o at any point of the sphere, it j has sometimes positive and sometimes negative values. Thus, in the case at hand, if the potential function is not equal to o at all points in the space outside the surface ï, it cannot have values of the same sign at all these points; o can then be neither a lower limit nor an upper limit for it; its lower limit and its upper limit are both reached at points on the surface T. If we compare this result with the one we arrived at in the case where the potential function is equal to o at any point on the surface T, we can easily see that we can state this general theorem: When a closed area contains all the acting masses in its interior or on its surface and the sum of these l48 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. is equal to o, the values of the potential function at V outside this surface are bounded above and below by two of the values of this function on the surface; these two bounds are of opposite sign, unless they are both equal to o. CIIAP. rv. - THEOREMS ON THE SIGN OF THE ELECTRIC DENSITY. 1 49 CHAPTER IV. SOME THEOREMS ABOUT THE SIGN OF THE ELECTRIC DENSITY AT THE SURFACE OF A CONDUCTOR. The theorems demonstrated in the previous chapter allow us to prove some very simple and general propositions about the electrical distribution on the surface of a system containing exclusively conducting bodies. When the electric density has the same sign at all points of the surface of a conductor, we will say, with M. Cari Neumann, that the distribution is monogenic at the surface of this conductor; when, on the contrary, the electric density has different signs at the various points of the surface of a conductor, we will say that the distribution is amphigenic. Theorem L - -5*/ a system is formed by a single electrified conductor, the distribution on the surface of this conductor is necessarily monogenous. Let A be the constant value of the potential function at the surface of this conductor, a value which we suppose to be different from o. Let us imagine, to fix the ideas, that it is positive. In the field outside the conductor, we know that the potential function will take values all between o and A, therefore all lower than A. We will have then, at any point on the conductor, ,;^ << o, and, therefore, o- >" o. If we had assumed A to be negative, we would have had, at any point on the conductor, u <^ o. If we had assumed A equal to o, we would have had, at any point on the surface of the conductor, !7 = o. The stated theorem is thus proved. l5o BOOK II. - DISTRIBUTION AND PROBLEM OF DIRICIILET. This theorem is due to Gauss ('). The following ones are due to M. Cari Neumann (-). Theorem II. - Two conductors C and C, carrying cjuelconque charges, are put in presence and subtracted from V influence of any other electrified body; the distribution is monogenous at least on one of them. Let V be the potential function; let A and A' be the potential levels of the two conductors C and C, i.e. the constant values at which the potential function inside these conductors is reduced. If we consider the set of points outside the two conductors C and C, the values of the potential function at these points are limited below and above by two of the three quantities G, A, A'. So, in any case, at least one of the two quantities A and A' is either the lower or the upper limit of the values that the potential function takes in the cliamp. Let us suppose, for example, that the quantity A plays this role of limit. It will be easy, by repeating a proof analogous to that of the previous theorem, to prove that the distribution on the surface of the conductor (^ is necessarily monogenic. The preceding theorem is entirely general. Other similar propositions can be given, which are useful when one has some information either about the charges of the conductors involved, or about the potential levels. Let us first look at the theorems whose application requires some information about the charges of the conductors. Theorem III. - In the presence of an armature conductor containing a total charge equal to o, we place an electrified inductor; the distribution on the armature being necessarily (') Gauss, Allgerneine Lehrsàtze. . . arl. 28 {Gauss Werke, Bil V, p. 33 1). (') C. Neumann, Untersuchungen ûber das logarithmische iind Newton'sclie Potential, Chap III, §§ 4, 5, 6. Leipzig; 1877. CIIAP. IV. - TILEONS ON THE SIGN OF ELECTRIC DENSITY. l5l amphigcne, the distribution on V inductor will necessarily be monogenic. This theorem follows immediately from the previous one. TnÉouiiME IV. - If V one puts in presence two conductors carrying Vun the total load + pi, Vautre the total load - [JL, the distribution on each d^eux is necessarily mono se ne. Indeed, according to the theorem proved at the end of the previous chapter, the upper and lower limits of the values of the potential function in the field are then the potential levels of the two conductors. A proof similar to that of Theorem I shows that each of them carries a monogenous distrihution. TNEIiiiME ^. - A charge - i, concentrated at any point, produces a monogenous distribution on a conductor carrying the total charge -\-. This theorem is a special case of the previous one. Theorem \1. - The charge - i, concentrated in any point V., produces a monogenous distribution on a conductor carrying the total charge (i + [x), a being any positive quantity. Let A be the potential level of the conductor G. The point P can be considered as a fading surface at any point atc which the potential function has the value - oo. The limits of the values of the potential function in the field are thus two of the three quantities - yz, o, A. - 00 is necessarily the lower limit of the values of the potential length in the field. The upper limit of these values cannot be o, because the potential function would have negative values at any point of the field, while it is easy to see that at a point very far from the acting masses it reduces significantly i52 IJVRE II. - DISTRIBUTION AND DIRICHLET PROBLEM. to -, i.e. to a positive quantity. Thus the upper limit of the values of the potential function is A, and it follows immediately that the distribution on the conductor G is monogenous. Let us now consider the proposals for the cases where the two conductors involved, C and C, are held at known potential levels. Tneorium VII. - If two conductors in presence are maintained at potential levels of the same sign, the distribution is m.onogenous on the one of the two whose potential level is greater in absolute value; it has the same sign as the potential level. If, for example, we have A>A >o, A will be the upper limit of the values of the potential function in the field, and then, on the conductor C, the distribution will be monogenous and positive, which demonstrates the stated theorem. Theorem VIII. - If two conductors in presence are maintained at potential levels of opposite sign, on each of them the distribution is monogenous and of the same sign as the potential level. The two limits between which all the values of the potential function in the field are computed are, in fact, in this case, A and A'. Theorem IX. - If one of the two conductors, C, is kept at the potential level A, and the other, C, at the potential level o, the distribution is monogenous on each of them; on C, it has the sign of A; on C, a sign opposite to that of A.. In this case, in fact, the two limits between which all values of the potential function in the field are included are A and o. CIIAP. IV. - THEORIES ABOUT THE SIGN OF ELECTRIC DENSITY. l53 Theorem X. - Sa/- a conductor put in communication with the ground, an electric charge of a certain sign, concentrated in a point, generates a monogenic distribution of opposite sign. This is a special case of the previous theorem. Let us note, in conclusion, that Theorem I can be generalized. A proof analogous to the one that provided this theorem will establish the following one: TneoRiiM XL - In a system formed by any number of conductors, at least one of these conductors carries a monogenous distribution. All the demonstrations we have just given assume that the surfaces of the conductors bound a linearly connected space indefinitely extended in all directions. The correctness of the previous theorems therefore assumes that all conductors are external to each other. They may cease to be applicable if one of the conductors is inside a cavity whose walls are formed by the other. It is to this very important case of electrical distribution that Chapter V of Book 111 will be devoted. 154 BOOK II. - DISTUIIJUTION AND PROBLEM UE DIUICIILET. CHAPTER Y. THE PROBLEM OF LEJEUNE-DIRICHLET. § 1 - The problem of electrical distribution can be reduced to the Lejeune-Dirichlet problem. The general problem of electrical distribution can be stated as follows: A number of bad conductors A, A', A" . . carry certain distributions. A number of conducting bodies B, B', B" . . are insulated and carry certain total electric charges Q, Q', A number of conductive bodies C, (7, i"' . . .) are maintained at certain potential levels y, v', v" . Is there an equilibrium electrical distribution on conductors B, B', B", ... . C, C, C", ... and what is this distribution? Let us first recall that, according to the proof of M. Liouville, which we indicated in Chapter II, a similar problem, if it admits a solution, admits only one. This theorem is essential for the study that follows. The problem we have just stated can be broken down into two successive problems, each of which must admit at most one solution according to what we have just said. The first problem is this: Find the electrical distribution generated by bad conductors A, A', A", ... on good conductors B, B', B", ..., C, (Z' , C", held at given potential levels b, b', b", ..., c, c', c", ..., whatever these potential levels are. This first problem being solved, we will have to solve the next one. In the previous distribution, the respective total loads of the CHAP. V. - LK PROBLEM OF THE YOUNG-DIRICIILET. l5i drivers B, B', B", . . . are functions of the quantities b, V , b". . . c, c', c", . . . which can be calculated when the distribution is determined. Let q {b, b', h". q'{b, b". q'{b, b". . c,c ,c . c, c', c" . c, c', c" these expenses. We will have to determine the quantities b, b', b", . . the equations q {b,h', b". ... c, c',c", ...) r^ Q, q'(b,b', b"., ... c,c',c", ...) r^ Q', q"{b, b'. b". . . c, c', c". . . . ) = Q", Cj C j C , pa It will then suffice to transfer these values of b, b', b", ... c, c', c", . . . in the solution of the first problem to have completely solved the question of electrical distribution that we had asked ourselves. The difficulty in studying electrical distribution usually lies in solving the first problem; the second is of lesser difficulty. Let's see, for example, how the second problem can be solved in the case where the system contains only one conductor. If this conductor is the conductor C, maintained at the potential level y, the equations of the second problem reduce to the fully solved equation c := v. If this conductor is the conductor B, carrying a given charge Q, the second problem comes down to solving an equation of the first degree. Let's imagine, for a moment, that we have solved the following problem: Since the conductor B is not influenced, how must V electricity be distributed to its surface to maintain the potential level i? Suppose that S is, in this problem, the electric density at each point of the conductor B and ^the charge distributed on its surface. l56 BOOK II. DISTRIBUTION AND PROBLEM OF DIRICIILET. On the other hand, in the first problem, soil '^(0) is the electric density at any point of the conductor b. If we place in the presence of the nonconducting body A the conductor B, and if, at every point on its surface, we place electricity with density t(6) + 6'S, we shall obviously have a new solution of the first problem, a solution in which the potential level of conductor B will be (^ + b'). The total charge of this conductor will have become q{b) -{- b' ■^. We have then The function q{b) is therefore a linear function of 6, and the equation to solve reduces to a first degree equation in b. It is therefore on the solution of the first problem that our attention must be focused. The conductors B, B', B", . C, C", C", ... can be hollowed out by cavities^ some of them can be contained inside the cavities of which others form the walls. The space not occupied by the material of these conductors can thus be broken down into several linearly related spaces. One of these spaces extends to infinity, the others are limited on all sides by the walls of the conductors. Let V be the potential function, at a point in space, of all the charges distributed on the system; let U be the potential function of the charges distributed on the bad conductors; let W be the potential function of the charges distributed on the conductors. We have V = U -+- W. The function U is, moreover, a given function, so that the determination of the function V, which is sufficient to solve the problem of electrical distribution, is reduced to the determination of the function W. Let us see what conditions this function W is subject to. Consider the linearly connected space outside all conductors. In this space, the function W is harmonic. If R is the distance from a point to the origin of the coordinates, t^W t)W dW àx oy dz CHAP. V. - THE PROBLEM OF DK LEJEUNE-DIRICHLET. I Sj remain finite quantities when R grows beyond the limit. The space in question is based on certain surfaces S, S', S", . T, T', T", . . belonging to the conductors H, B', B", . ., C, C, C", .... On the surface S, we have W - b - U, On the surface S', we have VV - b' - U. On the surface S", we have W = b" - U, On the surface T, we have W= c - U, the quantities [b - U), (h' - U), {b" - U), . . ., (c - U), . . . being data of the problem. Now consider a linearly connected space liniil<* on all sides by the conductors. Let s, s', s". ...,/,;', l", . . . be the surfaces belonging to conductors B, B', IV', . .C, C, C". . . which limit this space. In the considered space, the function W is harmonic On the surface s, we have W ~ b - L, On the surface s', we have W - Z*' - U, On the surface /, we have W = c - V. Conversely, if we have determined a function W which, in the various spaces considered, satisfies the conditions we have just indicated, we have determined the function we are looking for. Indeed, suppose that, in the various linearly related spaces not occupied by the conductors, we have determined a function W that verifies the previous conditions. To obtain a function W defined in the whole space, we will agree to give to this function the value b inside the conductor B, the values b', b", . . . ... inside the conductors B',B",...,C,C',C",.... Then the function W will be finite, uniform, and continuous throughout the space; it will be harmonic within each of the linearly related regions into which the unbounded space is partitioned<' 1)8 BOOK II. - DISTRIBUTION AND PROBLEMS OF DIRICIILET. by the surfaces that separate the conductors from the insulating medium; it will take values given on each of the surfaces in question. When R grows beyond any limit, the quantities RW, R--t- > R2-- -, R* -- will keep finite values. According to what we have VU in Book 1, Cli. Vil, § 5, only one function W can have this set of characters; the function W we found therefore coincides with the one we were looking for. The problem of electrical distribution is thus reduced to the resolution of a certain number of problems reducible to two main types: these two types, to which we will give the names of external problem of Lej euiie-Dirichlet and internal problem of Lejeune-Dirichlet, are characterized as follows: i" PnoBi.iiMK KXTERIOR OF Lkjkitve-Dirichlkï. - As a linearly connected, unbounded space, external to some closed surfaces S, S', ... find a function V, harmonic at any point of this space; equal to o at infinity, so that nue RV, R- - -> R- -- > R--t- keep finite values ^ ' ox dy Oz '=' ■' when the distance R from the point [x,y^z) to V origin of the coordinates grows beyond any limit; and taking on the surfaces S, S', ... given values which vary in a continuous way from a point to Vautre of these surfaces. 2" Lejeuke-Dirichlet's PROBLEM 1^TER1ELR. - Given a linearly connected space, bounded by some closed surfaces S, S', ... find a function , harmonic at any point of this space and taking on the surfaces S, S', ... given values which vary in a continuous way from one point to another of these suif aces. The solution of any electrical distribution problem involves the solution of an external problem and a number of internal problems. In the particular case where the system consists exclusively of solid conductors outside each other, the electrical distribution on this system is determined by the solution of the external problem alone. CHAP. V. - THE PROBLEM OF LEJELNE-DIRICHLET. 1 Sq .^ 2. an attempt to demonstrate the principle of Lejeune-Dirichl and. It was Riemann who, in 185^ C), gave the name of Dirichlet problems to the preceding problems, and of Dirichlet principle to the statement which affirms, in all cases, the existence of a solution for these problems. This name, now adopted by geometers, is a very bad choice; for Lejcune-Dirichlet has never published anything on this subject. This problem is, in reality, as we shall see in the next chapter, only another form of a problem posed by Green in 1828. Gauss (-), in iSSg, also formulated a problem equivalent to this one, and proposed a demonstration tending to prove that it always admits a solution. Finally, in 1847, Sir W. Thomson (3) stated, and even generalized, the so-called Dirichlet problem, and, by a method similar to that of Gauss, showed that it always admits a solution. The demonstration proposed by Gauss served as a model for the other demonstrations that have been proposed to prove the Dirichlet principle, that is, the existence of a function that solves, in all possible cases, one or other of the Dirichlet problems. Gauss's demonstration was taken up by M. E. Mathieu (' ). The two demonstrations proposed by Sir W. Thomson obviously derive from the same idea. The same is true of the one we are about to read. This one is found, for the first time, restricted to the case of two variables, it is true, in the Memoir of Riemann that we have quoted; Nalani ("*) has exposed it first for the case of three variables (') li. l\iKMAXX, Théorie cler Abel'schen Funktionen {Borchardt's Journal, t. LIV, p. ii5; 1857. - Biemann's gesammelle Werke, p. 90). {') Gauss, Allgemeine Lehrsiitze. .., art. 31 {Gauss Werke, Bd V, p. a'il). - We shall return to Gauss' problem in I^ivre III, Cliap. V. (') W. Thomson, Theorems with reference to the solution of certain partial differential equations ( Cambridge and Dublin niatheniatical Journal; January iS'iS. - Translated into French in Liouville's Journal, t. XII, p. 49-3; 1847. W. Thomson, Reprint of papers on Electrostatics and Magnetism, n" 206). (*) Mathieu, Réflexions au sujet d'un théorème d'un Mémoire de Gauss sur le potentiel {Borchardt's Journal, t. LXXXV, p. 264; 1878). (^) Natani, Mathematisches Wôrterbuch, Bd V, p. 602; 1866. l6o BOOK II. - DISTRIBUTION AND PROBLEM OF DIRICIILET. bles, but one and the other declare that they borrowed it from the teaching of Lejeune-Dirichlet. We find it, in fact, in the Lectures of Lejeune-Dirichlet, written by M. Grube (*). This demonstration, as we shall see when we explain it, gives rise to several very serious criticisms, some of which also affect Gauss's demonstration and all those which derive from it. These criticisms are due to M. Weierstrass, to M. Kronecker (2) and to M. Heine (3). We will demonstrate, by the method of Lejeune-Dirichlet, the existence of a solution for the interior problem; one would see without difficulty that one can construct, for the exterior problem, an analogous demonstration subject to the same criticism. Let then be a closed linearly connected space bounded by a<; closed surface S which can be composed of several parts. Let q be a continuously variable value on the surface S. Let us assume, first of all, that there exist an infinite number of functions Q, regular in all the space considered, and whose value at the point (i^x^y, z) tends to the value ^when the point (yXj y, z) tends in some way to a point of the surface S. This supposition, which Dirichlet considers as obviously permissible, might well not be legitimate. M. Heine has shown, it is true, that if there is such a function Q, there is certainly an infinite number of them; but he has pointed out that the existence of such a function in all possible cases is by no means certain. (') P.-G. Lejeune Dirichlet, Vorlesungen ûber die in umgekehrten Verhàltniss des Quadrats der Entfernung wirkenden Kràfte, lierausgcgebea von F. Grube, p. 127. Leipzig; 1867. (^) Mr. Weiersirass and Mr. Kronecker have not formulated these criticisms in any of their publications; but Mr. Heine, Ueber trigonometrische Reihen {Borchardt's Journal, Bd LXXI, p. 36o; 1870) and M. Bruns, Deproprietate quddam functionis potentialis corporuin homogeneorum {Inaugural Dissertation,^. la; Berlin, 1871), in making them known, declare that they have borrowed from the teaching of the two illustrious geometers. (' ) Heine, Loc. cit. and Ueber einige Vorausselzungen beini Beweise des Dirichlet'schen Princips {Matheniatische Annalen, vol. IV, p. 62G; 1871). CH.\P. V. - THE PROBLEM OF LEJEUNE-DIRICHLET. l6l Having made this first assumption, let us consider the integral extended to our enclosed space. This integral is finite and never negative; its values therefore admit a lower limit. Dirichlet takes it for granted that there is at least one determination of the function Q which makes J take this limit value. Mr. Kronecker and Mr. Weierstrass have rightly pointed out that a quantity whose values are finite does not necessarily take its limit value, so that the demonstration here presents a second doubtful point. Nevertheless, let us assume that, among the functions Q, there is at least one, V, which makes J minimum. We will prove that this function verifies, at any point inside the considered space, the equation AV = o. It will then be proved that the function V, which takes on the surface S the values ^, solves the Dirichlet problem inside the considered space. The function V being regular, having consequently partial derivatives of the first and second order which are continuous, the quantity AV varies in a continuous way inside the considered space. We can therefore always divide this space into a certain number of regions which will be of three kinds: 1° Regions A, A'. ..., where AV is equal to o; 2° Regions B, B'. ..., where AV is positive; 3° Regions C, C, ..., where AV is negative. These regions are bounded by the surface S and by surfaces along which AV = o. Let us imagine that we form a function T, regular in the considered space, taking at any point of the surface S the value o; positive at any point of the regions B, B', ...; negative at any point of the regions C, C, ...; equal to o, consequently, on the surfaces which separate these two kinds of regions; any inD. - I. II 102 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. end in the regions A, A', .... Dirîchlet admits the existence of such a function, but this existence is not obvious. This is the third weak point of his demonstration. Let h be an arbitrary constant. The function U = Vh-AT is one of the functions Q, whatever the constant h is. For this function U, the integral J becomes J J J \dx dx dy dy '^ Oz Oz ) ^ From what has been said, the function V makes the integral J minimum. So the quantity J is now a function of the second degree of li which must be minimum for h=.o. For this to be the case, it is necessary to have J J J \ ^x dx toy dy dz dz / J ^ - - The function T is equal to o at any point on the surface S. Therefore, Green's identity allows [Liv. I, Chap. III, equality (3)] to replace this condition by the following one: r r CtW dx dy dz = o. The product TAV, zero at any point of the regions A, A', ..., is positive at any point of the regions B, B', ..., C, G', .... The previous equality can only exist if the regions B, B', ..., G, G', ... do not exist. We have then, in any point of the considered space, AV = o, and the function V solves the Dirichlet problem. We have seen how unsatisfactory the demonstration of the Dirichlet principle, given [by Lejeune-Dirichlet, is, although it has the advantage of giving a new and interesting point of view to the problem at hand. We will see, in the following Ghapitres CHAP. V. - THE PENOBLEM OF LEJEUNE-DIRICHLET. l63 It is clear that Ton can demonstrate Dirichlet's principle in cases which, in spite of their extent, remain particular. When admitting the correctness of Dirichlet's principle in the most general case, we must not forget that we are making an assumption whose verification is beyond the scope of the present analysis. § 3 - Statement of a more general problem than that of Lejeune Dirichlet. Since the electrical equilibrium on a system of conductors must be unique, each of the Lejeune-Dirichlet problems must admit at most one solution. It is easy to ensure directly that this is the case, not only for the Lejeune-Dirichlet problem, but also for a more general problem, which can be stated as follows: We give a closed space, bounded by a closed surface S. We ask to find a harmonic function V in any closed space of which it is a question, continuous, as well as its first derivatives, in any point of the surface S, when we know the values of V in any point of a part S' of the surface S and the values of -^ at any point of Vautre part S" of the surface S. Suppose that this problem can admit two distinct solutions, and be the difference of these two solutions. Then, in At any point in S', we would have = o, and at any point in S", -r^ = o, which would allow to write S or else, the function being necessarily harmonic in the considered space, l64 LIARE II. - DISTRIBUTION AND DIRICHLET PROBLEM. which would result in of _ of _ of _ dx ~ ' dy ' dz at any point of the considered enclosed space. The two solutions of the problem posed can therefore, within the space considered, differ only by a constant. Again, this constant is reduced to zero as long as V is given, even if only at one point of the surface S; for, at this point, we must have = o. We can state an analogous problem for the space outside the surface S, adding that at infinity V must have the characteristics of a potential function. We will show, as we have just done for the interior problem, that this exterior problem cannot admit more than one solution. CH.VP. VI. DK GREEN FUNCTION. l65 CHAPTER VI. THE GREEN FUNCTION. § 1 - Green's problem is equivalent to Dirichlet's problem. The so-called Dirichlet problem is equivalent to a problem posed by Green in 1828 (' ). This problem can be presented, from a logical point of view, if not from a historical point of view, as the search for a generalization of Gauss' mean theorem. Inside a sphere of radius R, a function V is harmonic. Its value Vo, at the center of the sphere, can be deduced from its values at the surface of this sphere by the relation (0 4^Vo= j^gvrfS, the integral extending to the surface of the sphere; this is the mean theorem. Equality (i) can be written in a slightly different form. Let r be the distance from any point in space to the center of the sphere. Let N be the normal to the element from to the interior of the sphere. At a point of the element dS, we have and EL 1 R2 d' d^i so that equality (i) can be written (2) 4TrVo r S^5n;.''s (') George Green, An essay on the application of mathematical Analysis to the theories of electricity and magnetisni; Noltingham, 1828, Art. 5 {Mathematical papers of the late George Green, edited by Ferrer s, p. 3i). l66 BOOK H. - DISTRIBUTION AND DIRICIILET PROBLEM. Green's problem aims at generalizing this equality. Consider a closed, linearly connected space bounded by a closed surface S {fig- 28), this simple connection or Fii?. 23. multiple, and a fixed point Mq (.^0,^05 ^0) inside this space. Let M(.r, y, z) be a variable point of the same space, and let r represent the length MMo We ask to find a function G{x,y, :z) that verifies the following conditions: 1° This function is harmonic in all the space considered, except at point Mq; 2** At the Mo point, it is infinite, but the difference G(;r, jK, z) is harmonic in all V space, even at point Mo ; 3" Any function V, harmonic in V considered space, takes, at point Mo, a value Vq given by V equality (3) ^--s^-^ ^s. This is Green's problem; the function G, which this problem aims to determine, received from Riemann the name of Green's function. The comparison of egalities (2) and (3) shows that, in the particular case where the surface S is a sphere, and where the point Mq is CHAP. M. - THE FUNCTION OF GREEN. I67 the center, we have r The mean theorem determines the Green's function in this case. It is easy to see first that Green's problem cannot admit, for a given surface S and for a given position of the point Mq, more than one solution. Let us assume that, for the same surface S and for the same point Mo, we have found two Green's functions, (j[x,y,z) and G'(x,y,z). Then, whatever the harmonic function V is, designating by Vo its value at the point Mo, we can write dG 47rVo= J^ V and also f)G or, therefore. °=S^"-'^âN-^''^ ()(G - G') But both functions G (x^,/, z) - -, are, by hypothesis, harmonic functions inside the considered space. It will thus be the same for the function G(a7, jK, z) - G'{x,y, z), so that we can write S(G-G') ,à(G-C,') dNi dS = 0. The function (G - G') being harmonic, this equality can be transformed, by Green's identity, into and, by a process that we have followed several times, we deduce l68 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. of this equality that one must have, in any point of the considered space, O(G-G') ()(G - G') ()(G- G') - G, = Oi dx dy Oz The two functions G and G' can thus differ only by a constant - they do not provide two really distinct solutions of Green's problem, since the function G enters the equality (3) only by its first derivatives. This proposition shows us that, if we obtain, by any means, a solution of Green's problem, we will have obtained by the same fact the general solution. We will obtain a solution of Green's problem in the following way. Let r(^, y, z) be a function of x^ y, z, satisfying the following conditions: 1° At any point of V space considered, it is harmonic; 2° At any point on the surface S, it takes the value , The function (4) G(^,jK, :;) = r(^,jK, 5)-+-^ solves Green's problem for the surface S, and the point Mq. Let us consider, in fact, any function V, harmonic in the considered space; Green's theorem will give or, since the function F is equal to - - at any point on the surface S, On the other hand, from the point Mq as center, let us draw a spherical surface c with a radius R small enough so that this surface is all within the considered space. The function - will be, as the function V, harmonic in all the space included CHVP. VI. - L\ FUNCTION OF GREEN. 169 between the sphere s. This identity is due to Green (' ). It is of great importance: we will often have to use it. (' ) George Green, An Essay. . . , Art. 4 ( G. Green's Mathematical Papers, p. 29). 170 LIVRK II. - DISTRIBUTION AND PUOULÈMIÎ OF DIRICIIKET. The demonstration of the proposition we have stated results immediately from the comparison between this identity and the identity (a), a comparison which gives us t)(r-+- ' 47rVo = Q V -^-,- ^ dS. The proposition we have just demonstrated shows us that: If we know how to solve the Dirichlet problem for the region inside a certain surface S, we know how to determine the Green's function for the surface S and a cjuelconque pointMo inside this su/face. 11 It suffices to refer to the statement of Green's problem to see that, conversely, if we know how to determine Green's function for a closed surface S, whatever the position of the point Mo at U inside the surface S, we know how to solve Dirichlet's problem for the space inside this surface. What we have just said shows the exact equivalence of the inner Dii-ichlet problem with the problem we stated at the beginning of this paragraph, and which we will call the inner Green problem. In the same way, we can show the exact equivalence of the external Diricblet problem with the external Green problem, which we will state as follows: An unbounded linearly connected space is external to some closed surface S i^fig. 24)1 with simple connection or Fig. .4. W^çc.y, ■A/) multiple; we give a point /î^e Mo(^05 ^o? z-o): being part of this space. riIAP. VI. - THE FUNCTION OF GREEN. IJI Let M(;r, jKi ^) be a variable point in the same space and let r be the distance MMo We ask to deceive a function (j{x^y, z) that verifies the following conditions: i" The function G(:r, y, z) - - is harmonic in any the space considered. 2° When the distance R from point M to V origin of the coordinates grows beyond any limit, the quantities RG, R2 -- j R2 -- , R- -T- remain finite. ox oy oz ^ 3° Let be a harmonic function in V considered space, such that the quantities RV, R- - ■> R- -;- " R- -pi remain finite when R grows beyond any limit; let be Vo its value at point Mq. We have Nj being the normal to Vêlement dS towards the interior of the considered space. The external problem gives rise to the same remarks as the internal problem; only one difference is to be pointed out: while the solution of the internal problem is determined only to a constant, the solution of the external problem is entirely determined, since, at infinity, G takes the value o. § 2 - Fundamental property of the Green function. Riemann ('), who showed the equivalence of Green's problem and Dirichlet's problem, proved an important property of Green's function, a property that Green (-) had stated. Consider a su? face S (fig. 25); in its interior, two fixed points Mq(xo, yo, So), M'"(:r'", jk'o, -s'o) ^^ "'^ point (') RiÉMANN', Schwere, Elektricitàt und Magnetismus, bearbeitet von Hatten^Of'ffy P- 142; Hanover, 1876. (') G. Green, Essay..., Art. G {Green's Mathematical Papers, p. 36). 172 BOOK II. - DISTRIBUTION AND PROBLEMS: OF DIRICIILET. mobile M(^, y, z). Let G(.r, jk, z) be the Green's function which has its pole in Mo and G'(^, j', z) the Green's function which has its Fig. 25. pole in M'^. These two functions can be taken as (6) G(a7'o,yo,z;) = G'(^o, JKo, so). Let r and /■' be the distances from point M to point Mq and point M'(,. We can take G {oc, y, z)= - r {x, y, z), G'(^,7, -)= ^X{x,y, z), the two functions r and F' being harmonic at any point inside the surface S, and reciprocally equal to ■> -,i on the surface S; it is obviously sufficient to prove the identity {'obis) r(^o> Xo' -0) = r'(^o, JKo, -3o). Now we have, according to the definition of the functions G and G', 4 ^r {x, jk'o, z'o) = ^r -^ds, dG r 'S These equalities easily give 4Tt[r(a7'o, jKo, -0) - r;(a7o, jo, ^o)J ^■Kr'{xo,yo, ^0) = S r' dN, dS. {d) èl The functions F and F' being harmonic within the CHAP. VI. - L\ FUNCTION OF GREEN. 178 pace that limits the surface S, we have, according to Green's theorem, atV T' dS = G. These functions F and F' reduce to and ; on the surface S, equality (d) becomes {e) 4TT[r(a7'o,yo>-o)-r'(^o,ro,-o)] = Sg\p dN]~7 wj ^^' To calculate the second member, let's draw two spherical surfaces, T and 'i J J R' = 471 Mo M'/ The identity (/) becomes Ss\^5^""7-dN;y^^ = " 174 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. This result, carried over into the identity [cl), provides the identity (6 bis) that we wanted to prove. Identity (6) can be translated in another way. LetY(^, jKî ^1 ^'ly'i ^') be a function such that T(-^' J- -2, ^0, JKo, ^o) = r(a^, 7, z), Let's then pose [{x'-xy^+iy-yy+iz'-zn-^ 11 It is easy to see that we will have g(x, y, z, xo, yo, -o) = G{x,y, z), gix,y, z, x,y'ç,,z'f,) :- G'{T,y,z), and that, consequently, this function g of six variables defines all the Green's functions which have their pole inside the considered space. The identity (6) can be written ^(^ô,yo)-o> ^0, JKo, -So) = 5^(^0,70,-0, a?;, jk'o, z'q). In other words, it expresses that the function g is symmetric in X, y, z and a;', y', z'. §. 3 - Determination of the Green's function in some simple cases. Since Green's problem is rigorously equivalent to Dirichlet's problem, the solution of one is not easier than that of the other. However, the change of form imposed on the problem allows, in some cases, to immediately see the solution. We will give some simple examples. 1° Green's function for the sphere. Let us take a spherical surface {fig- 26) and a pole Mo inside the spherical surface. Let AB be the diameter that passes through Mq. On this diameter let us take the harmonic conjugate point M^ CHAI". VI. - L.V FUNCTION OF GREEX. 17^ of the point Mo with respect to A, B. Let r and /' be the distances of the point M(x, y, 5) from the points Mo, M'". Fis. 26. The sphere is, as we know, the locus of points M such that /-' " OJVl'o ' K being the radius of the sphere. Let's consider the function om; I R 7' It is a function of the coordinates x^ y, z of the point M, which is harmonic at any point inside the sphere and which, at any point of the spherical surface, takes the value ;- It represents thus the function that we have designated by r(^, j', z). The Green's function, for the case where the Mo pole is inside the sphere, has the value if om; I (7) Gi{x,y,z)=-- -^ -r Let us now assume that the pole Mo is outside the sphere and that M" is its harmonic conjugate with respect to the points AB. We will still have at any point of the sphere OM' and the Green's function will still be given by the formula (7). The Green's function being known in all circumstances for the sphere, we will know how to solve the Dirichlet problem as well 176 BOOK II. - DISTRIBUTION AND PROBLEM OF DIRICIILET. for the space outside the sphere than for the space inside. The solution of the Lejevine-Dirichlet problem for the sphere was indeed given at the origin of mathematical physics, by the works of Legendre and Laplace (' ). Let us take, to represent the position of a point in space, the geographical coordinates r, 9, cp. When the value of the function V is given on the surface and this function is harmonic inside the sphere, it is expressed inside the sphere by a series of the form V = Yo + /-Yi -H r^Y.-^. . . -i- /-"Y," -+-. . . When it is harmonic outside the sphere, it is expressed in the space outside the sphere by a series of the form V = -" + --1 -+- I2 -- H- -^ -+- . Y" being a homogeneous function of degree n of the quantities cos8, sinOcoscp, sinOsin^, containing (a/^-f-i) coefficients that are determined by the values of V on the surface of the sphere. We only briefly mention this result without insisting on the important properties of Laplace's Yn functions (- ) or spherical functions, functions which played a great role in Laplace's research on celestial mechanics and in ( ' ) This form of development, obtained by Laplace, is a very particular consequence of the following proposition: Any function Y, harmonic inside a connected space, can, in an infinite number of ways, inside this space, be put in the form P" being a polynomial of degree n in x, y, z, satisfying the equation AP" = 0. This beautiful theorem is due to M. Paul Painlevé [Paul Paiislevé, Sur les lignes singulières des fonctions analytiques ( Annales de la Faculté des Sciences de Toulouse, t. II, p. B.112; 1888)]. In the case of the development in Laplace functions, the polynomial P" is homogeneous in x, y, z. (^) V. Heine, Handbuch der Kugelfunktionen. 2" edition; Berlin, 1878. CHAP. VI. L.V FUNCTION OF GREEN. 177 he works of mathematical physics of Poisson, Gauss, Neumann, etc. -2° The Green's function for V space between two planes that intersect at an angle commensurable with tz. The function of Green is easily found for the interior space to a commensurable dihedral with you. Let us take, for example, the case of the dihedral equal to -; it will be easy to see that what we are about to say extends to the general case. Let AOB be the plane angle of the dihedral {fig. 27) and Mo the pole. Let us replace the two planes OA, OB by two mirrors, and Mo by a luminous point. Let N, N', N", Q, P, P', P , Q the two sequences of images provided by the point MoLet M be a variable point. If the point M is placed on the plane OA, we have MMo MN I MP I MP I 7 =0, MN' MN" I i _ MF' "" MQ ~ '^^ D. - I. 178 BOOK 11. - DISTRIBUTION KT PROBLlhlK OF DIRICHLET. If it is placed on the OB plane, we have I T _ mm;~ MP ^''' I I o, MN ■ MN' I I MP' "MF' I I The expression MM^ ~ VMN ~^ MP/ '^ \^Û^ ^ MN'/ ~ \MK' ~^ MF/ ^ MQ whose formation law is easy to generalize, is equal to o at any point of the two planes OA, OB. It can be seen that at any point on the surface which limits the dihedral the function ~ (, MW"^ "MP j"^MF"^ MN') " \MN'' "^ MF j"^ MQ is equal to -; moreover, it is harmonic within the dihedral, and, at infinity, behaves as a potential function. It thus represents the function r(x, jk, 5) and the Green's function is i*represented by (8) < ^ ^ ("^ \MP'^mF/ ViVW "^ MP'7 "^ MQ* 3" The Green's function for V space between two parallel planes. Let AA', BB' {fig- a8) be the two parallel planes between which point Mq lies. Let us replace these planes by two mirrors and the point Mo by a luminous point. Let a,, ^i be the images of point Mo with respect to mirrors AA', BB'; let ao, ag, ... be the successive images of point ai and ^25 \^3i --- the successive images of point j^j. Let us consider the series T / 1 T / 1 I ^ / I I MMo ~~\M^ "^ Mpj/ "^ VM^ ^ M^/ "" \M^z "^ Mp^ CH.\P. M. - THE FUNCTION OF GREEN. 179 This series with alternating terms whose general term tends to o is absolutely convergent. We can write it r I F I I I and, in this form, we see that it is equal to o when the point M Fis. 2i B' ^M X, B A lies on the plane AA'. It can also be written I I II I I mm; ~ M^i "" M^"^ M^"^ Mp^ ~ Mfs ~ - ■ - ' and, in this form, we see that it is equal to o when the point M is on the plane BB'. The quantity I I M^ I M^ .Ma, ■ MJii/ ' VMao ' M^^/ V^as "^ MpJ '^ * * " ' which is uniformly convergent and harmonic at any point in the space between the two planes, which behaves at infinity as a potential function, is equal to on each of the two planes. It is therefore the function T(cc,y, z) and the Green's function has the value (9) (G(.,^,.)=i-(jJ^ M pi Me MpJ" VMas "*"Mp3J"^ r80 BOOK II. - DISTRIBUTION KT PRORLÈME DE DIRICIILET. CHAPTER Yll. TRANSFORMATION OF THE EQUATION AV=: o INTO QUELCONICAL ORTHOGONAL COORDINATES - ELECTRICAL DISTRIBUTION ON AN ELLIPSOID. § 1 . - Transformation of the equation AV = o into orthogonal coordinates any. We have just seen how, in certain cases, one could see, a priori, the form of the Green function, and, consequently, solve the Dirichlet problem. But these are exceptional cases. A powerful method for finding the solution of the Dirichlet problem in a number of cases is to replace the rectangular Cartesian coordinates used to define a point in the space under consideration by suitably chosen orthogonal curvilinear coordinates. The partial differential equation AV = o is then transformed into another partial differential equation whose integration can be discovered much more easily. This method was created by Lamé (') who made great use of it in his work (2) and obtained magnificent results. After Lamé, Jacobi (^) must be mentioned as the first person to have treated a similar transformation in general. Let u^ v^ w be the curvilinear coordinates of a point with Cartesian coordinates x^ y, z. The change of coor (' ) Lamé, Mémoire sur les sur/aces isothermes... . {Journal de Liouville, t. II, p. 147; 1837). (^) See, especially, Lamé, Leçons sur les coordonnées curvilignes et leurs diverses applications; Paris, iSSg. (') Jacobi, Ueber eine particulare Losung der partiellen Differentialgleichung AV - {C relie' s Journal, Bd XXXVI, p. ii3). TRANSFORMATION OF THE EQUATION AV = O. is defined by the equations / X-f (UjV, w), (0 (give him / , dx , dx , dx , ax = '^- du + -~ dv -h -- dw, or ov dw (?0 from at dv ôw , dz j toz j toz , dz = -:- du -i- -- dv -h ^-- dtv, or ov dw and, therefore, dx^ -H dy^ -H dz'^ = (3) { dxy said) of) J rdv, w), Ji^u^ du,v, w -\- dw). C{u,v, w -T- dw), G'(w -h du, v ■+- dv, w). The lengths of the edges of this parallelepiped are given by the relations In fact, we have ( 5 ) mm'' = dx"^ -4- f/j2 + dz^. If the curvilinear coordinates considered are orthogonal, the small parallelepiped considered must be a rectangular parallelepiped. We must therefore have MM ' = MÂ^ -f- MB^ + MC' ; and this whatever the du, dv^ dw. If we relate this result to the equalities (3), (4) and (5), we see that, /?oi<7- the curvilinear coordinate system is orthogonal, we must have the relations dx dx dy dy dz dz dv dw dv dw dv dw ' ,", 1 dx dx dy dy dz dz dw du dw du dw du dx dx dy dy dz dz dv of dv of dv We will show that, conversely, if these relations (6) are verified, the curvilinear coordinate system is orthogonal. Let's say, following Lamé's notation, that TRANSFORMATION OF THE EQUATION AV = O. i83 Equality (3) will become ( 8 ) dx^ -^ dy^- -^ dz^- =^ II^ du"^ + H \ dv'^ -i- H | dw^. Let us agree that the quantities H, H,, Ho defined by the equalities ['j) are essentially positive, and write the equalities (2) in the form i dx ^j , i dx .. j I ^-^ Ti j dx = ^ -T- H rfjf -f- -p -^ Hi at^ -t- 77- -r- "2 dw, H du Hi dv H2 ow (9) \dy=^^-f\ldu---~^-fn,di>-^^n^dw, i dz i dz j dz dz - yj -^- ti du -\- T. ^ "1 dv -t- TT - ;: - "2 dw. \ H or lli ov Hj aw In these equalities (9), the coefficients of Hdii^ H, û?p, H2 âfw, verify, as shown in equalities (6) and (7), the same relations as the coefficients of a change of rectilinear and rectangular coordinates. H = Hj -r~ dx -{-ïîi-- dy -i ' dx toy -^ -H.g., .T J TI ^^'^ J TT ^"^ _7 TT à^ J H2 dw = H, -- dx -+- H, -7- dy -+- Ho -r- dz ; dx toy -^ dz the coefficients of this new transformation must still verify the same relations as the coefficients of a change of rectilinear and rectangular axes, so that instead of the relations (6) and (i^), we can write the relations (fi bis) ' tov dx div dx dv dw dv dy dy dz dw dw dx of dx dw of dw dy dy dz from Tz = ''' from . dx dv dx of the dv of dy dy dz dv I ^c duy /duy /'^"V dx) '^[dyj ^\dz) ' 1 Hf = = (: dvy / dvy dx) ^\ày) ' KS' I "2 -i: dwy /dw\' / dw\ (7 ^"") Let us apply these properties of the transformation of Cartesian coordinates into orthogonal curvilinear coordinates to the search for the form of the equation AV=-o in the new coordinate system. So that at any point in a space where the function V is regular, we have AV = o, it is necessary and sufficient that the integral TRANSFORMATION OF THE EQUATION AV = O. i85 extended to the surface that bounds any volume element of this space is equal to o. dY Let's write the equality dNi dS for the rectangular parallelepiped MABCA'B'C'M'. The surface element MBA'C has area M B A'C =: MB X MG = H, Hg dv dw. For this same element, the N/ direction coincides with MA. We have so dY_ _ V (A) - V(M) Wi ~ MA By the waySj V(A) - V(M)= ^^du. du MA = H du. We have therefore, for the element MBA'C, -- - db = - ,-f r- du dw. dNi H of For the element M'B'AC, we have dNi L H du du \ H du/ J These two elements will therefore provide the sum considered with the term d /HiH, dY\ j -, of dv dw . of the W of the We will thus find the equality o dN,- Ldi< V H du) dv \ Hj t^t^ / dw \ H2 (ii, (>, w) = o. If we replace x^ y, z with their expressions in w, v^ (P, the function /"(^, y-, z) turns into a function ^{ii, v, w). If we know how to find a function of u, f , w which verifies the equation (lo) in the space bounded by the surface ^(u, i>, w) = o, and which, on this surface, takes the same values as the function ç(",p, pp); it will be enough, in this function, to replace u, ç, w by their expressions in function of x, y^ z^ to obtain the sought function. We have said that this transformation of the Dirichlet problem has proved to be extremely fruitful in terms of results. § 2 - Transformation of the equation AV = o into geographic and elliptical coordinates. As an example, we will apply what we have just said to the transformation of the equation AV = o into two particular orthogonal coordinate systems. Let us first consider the geographic coordinates p, 9, o. The transformation formulas are X = Ç) cos6, y =^ p sinô coscf, ^ = p sinG sin o. TRANSFORMATION OF THE AVERAGE = O. 187 We then have "i=(:-f)'-(i)-(i)'=p-'"" Equation (lo) thus becomes ^") d-p (p'^'"^ d^j -^ dô (^''^^ âôj -" 5^ (ii^ -d^) = ^' or, by putting CCS 6 = [JL, This equation, transformed from the equation AV=o, is due to Laplace; it is even in this form that Laplace (') first made known the equation that bears his name. This equation is the foundation of the whole theory of Laplace functions, which allow, by serial developments, to solve Dirichlet's problem for each of the two spaces limited by a spherical surface. Let us now consider the case of elliptical coordinates. The functions u, v, w are, in this case, the three roots of the equation (12) - ^ -^ TT T + T 1 = 0, which we have already considered [Book I, Chap. VI, § 2]. We know that this equation has three real roots, separating the quantities We will take a-^. For a the root between -}- a^ and -f- oo ; For (V the root between - b^ and a^, which is negative; For ç the root between - oo and - b'-. (') Laplace, Théorie des attractions des sphéroïdes et de la figure des planètes {Mémoires de l'Académie des Sciences pour 1782; Paris, 1785). DISTRIBUTION AND MANAGEMENT PROBLEMS. The surfaces u = const. are ellipsoids; The surfaces v = const. are hjperboloVdes to a sheet; The surfaces (v= const. are two-sheet hjperboloids. All these surfaces have the same center, the same axis directions and the same foci; we know that they form three families of orthogonal surfaces. Let us calculate, for such a system, the Lamé coefficients H, Hi, Ho. DifTerentiate equation (12) with respect to x, by j replacing successively), by u^ (^, w. We will have of dx yy^aP- -\- u)'- ' (lj--h u)- u^ |_("--+-p)- {b'^-^vy- V- \ dx r a?2 JJ/-2 ^2 I (j,p L(a2-i- (pj2 ~^ (b--{- wy ' (v2 J ()^. "" ■iX The diierentiation with respect to ky or with respect to z provides two groups of analogous relations. From these relations, we can easily deduce Ui \dxj ^ {âj^J ^ [to. I __ /(Jtvy /d(v\- / dw H2 \dx J \dy J \dz X + yl , -S^ {b-i-i-uy"^"2 4 (a-4- uy x"^ -i y2 ^-2 (a^-h vy 4 (a2_}_(v)2 (^è24_^j,)2 ",2 These expressions of H, H,, Ho contain both the variables M, ç, w and the variables x, y, ^, whereas we need to obtain the expressions of H, H<, H2 in terms of the quantities u, v, w alone. Note, for this purpose, that we can write x^- y^- z"- ("_X)(p_X)(h'- )0. a2_i-X 62-)- X ' X (a2-i-X)(62-i-X)X ' take the derivative of this expression with respect to X and do TRANSFORMATION OF THE EQUATION AV = O. 189 k= a; we will have (.3) (a^-T-u)- (ô^H- a)^ if^ {a^-^- u){b^-\- u)u The first of the previous equalities then becomes the first of the equalities JJ2 _ (V - U)(W - II) (.4) {Hf = We will thus have 4"(a2-i- u)(62-H u) (iv - v)(u - v) (u - iv)(v - w) 4 "v ( a- -h w j ( 6^ -+- w } I / HiH2 y_ (v - wy- u(a^-\-u)(b^--hu) l V II / ~ 4 i>{a'-i-i>){b^-{-i;)w{a--^-h w){b^-\- w)' ^^-^^ '\ \ Hi / ~ 4 w(a^-+-w){b^-+-w)u{a^-^u){b-^-+-u)' HHi\2 ("_p)2 ,p( \ Hi (^(^ / (Jw \ H2 dw / give the transform of the equation A\ =0 in elliptical coordinates. § 3 - Electrical distribution on an ellipsoid removed from any influence. The transformation that we have just indicated will allow us to determine quite easily the distribution of a certain electric charge on an isolated ellipsoid and removed from any influence. Let us imagine that the equation of this ellipsoid is, in Cartesian coordinates igO LIVRK II. - DISTRIBUTION AND PUOBLIiiM OF DIRICIILET. OR, in elliptical coordinates, {i6 bis) u = c'-. Let us imagine that we find a function of the only variable u, tinie, continuous and uniform at any point outside the ellipsoid, equal to o at infinity, and verifying, at any point inside the ellipsoid, the equation (lo). This function, depending only on ", will take a certain constant value on the ellipsoid, and, consequently, will be the potential function of a certain quantity of electricity distributed on the surface of the ellipsoid subtracted from any influence. However, when the function V depends only on the variable m. the equation (lo) is reduced to the form from 11 from ) ~ ^' or, according to the first of the equalities (i5), / - ; - ; - ,-;; " y u(a'-i- u)(o^-i- u) -y- = - L, from C being a certain constant whose value is related to the quantity of electricity distributed on the ellipsoid, and the radical being taken in absolute value. As V must be equal to o at infinity, we have .-, r"^ of the (17) ){b-^-\-u) This is the value of V at the point of coordinates (w, ç, (v) outside the ellipsoid. The value of the potential function at any point on the surface or inside the ellipsoid is obtained by giving m, in this expression, the value c^ which corresponds to the points on the surface of the ellipsoid. If we denote by A this value of the potential function, we will have (18) A=Gr-=£^=. J^. \/u{a^-Jru){b^-\-u) The surface electrical density o- can be obtained by the formula i^ dV TRANSFORMATION OF THE EQUATION AV = O. I<)I The direction N^ of the exterior normal to the ellipsoid is the direction of the tangent to the line v = const., w = const., in the sense that the quantity u is increasing. Thus we have W^" n of U having the value c-, and, therefore. C (19) ^ = :r 2 c^ - *')(^' - ''^) This density can be expressed in Cartesian coordinates. If we make u = c^ in equality (i3), we find and equality (19) becomes r: ( ig bis) G This formula has an interesting interpretation. The tangent plane at the point oc, y, z k the ellipsoid is represented by the equation ^X vY zZ _ and the distance from the center to this plane is given by / y^ The formula (19 bis) thus highlights this fundamental theorem. When V electricity is distributed in equilibrium on the surface of an isolated ellipsoid, the surface density at each point is proportional to the distance from the center to the plane of the ellipsoid. 192 BOOK II. - DISTRIBUTION AND PROBLEM OF DIRICULET. Another interpretation of the formula (19 bis) can be given. Consider the ellipsoid E {fig. "too) and a second ellipsoid E', infinitely close to the ellipsoid E, concentric and homothetic to the ellipsoid E. Let k be the homothety ratio. Let M be a point on the ellipsoid E; let OP be the distance from the center O to the tangent plane at M to the ellipsoid E'. The ray vector OM meets the ellipsoid E' at a point M' whose M'D is the distance to the tangent plane considered. We have obviously MJ3 _ T - A6Y ~ ~li But M'D differs only by an infinitesimal amount from the thickness e at the point M of the layer between the two ellipsoids E and E'. There is thus proportionality between e and OP, and one can say that When V electricity is distributed in equilibrium on the surface of an isolated ellipsoid and subtracted from any influence, the surface density is at each point proportional to V thickness of the layer between this ellipsoid and a concentric ellipsoid, ]10 mo tic, infinitely close. In this form, this theorem was known by Poisson (' ). (') Poisson, Mémoire sur l'attraction d'un ellipsoïde homogène, read at the Acatléiuie on October 7, i833 (Mémoires de l'Académie des Sciences, t. XIII, p. 497; i835). CHAP. VU. - XnANSFORMATIO-N OF THE AV - O. IqS EQUATIOX Let's go back to the general formula from ,.7) v=cr-^=^ {b'--hu) and we propose to find the relation that exists between the constant C and the total electric charge Q distributed on the ellipsoid. We can do this simply by the following remark. Let's put R2 = .r^ -t- J^'^ -r- z^, and we can obviously write V Q " ^ -ÏÏ'^R^' a remaining finite when R grows beyond any limit. Consider the equation ;r2 r2 ^2 a^-h u b'^-r- u u ' and make it grow beyond any limit, we will see without difficulty that tends to i when u and R grow beyond any limit, so that we can write ¥=4-1-^, p remaining finite when u grows beyond any limit. On the other hand, we can write equality (17) in the form J " u^ of 8' 9.C fi' ^' remaining finite when u grows beyond any limit. By identifying these two expressions of V, we find the relation (20) Q = - 2C, which determines the constant C when the total charge Q of the ellipsoid is known. This relation allows us to determine the capacity K of the elD. - 1. i3 194 BOOK H. - DISTRIBUTION AND DIRICHLET PROBLEM. lipsoidc, i.e. the total charge that must be distributed on this ellipsoid to give it the potential level -- In this case, indeed, the relation (i8) gives i=cr '^ - ,, and the relation (20) gives These equalities, combined together, give (...) K = I'- ["( from § 4 - Special cases. The potential function of electricity spread over an ellipsoid is expressed, according to equality (17), by an elliptic integral of the second kind. There are two cases where this integral is reduced to the inverse functions of exponentials. These two cases are : i" The case where we sl c^b. The ellipsoid is then an elongated ellipsoid of revolution. a° The case where we have. a = c. The ellipsoid is then a flattened ellipsoid of revolution. 1° Ellipsoid of elongated revolation. Formula (17) gives of Q , \/â^= - - log^ ,r r'" thread . , (22) ^ = ~ -^- = = .-^'0g ^ ^H u s a' The capacity of the ellipsoid will be (3.3) K = Ja- -f- c elog v/a^î -I- c- -h a a° Flattened ellipsoid of revolution. The formula (17) gives here = 5 r, 2 J" from Let u = ^2, and we will reduce this integral to CIIAP. VII. - TRANSFORMATION DK l/EQUATION AV = G. I9S a form that can be integrated immediately; we will find arc tans ia a The capacity of the ellipsoid has the value r 7: I cl arc tane - L 2 a a " a J If T is the electric density at a point, this density is given by the equality (19), which becomes 47îQ 3" Flat circle. If, in the previous formulas, we make c = o, we obtain the formulas that refer to a plane circle of rajon a. In this z^ - x"^ -4- y2 case, - must be replaced by i '■ - ^ - , and we find arc tan!:!;^^ - it-]) = ^ ( 28 ) K = - , ( 20 I - = - ^ Q Jcf^ - {x'^-\- y^). ^ . vil. - TRANSFORMATION OF THE EQUATION AV = O. I97 our layer of thickness United in n infinitely thin layers. Black infinitely thin cone cuts respectively on the surfaces S,. So, . . ., S"_, elements A,B,, AoBo, . . ., A"_iBrt_|. Fig. 32. \>^y The small volume ABAi B( exerts at point M an action F| = sp AAi ofw. The small volume A, B, AoBo exerts at point M an action Fj = sp Al A2 dià. It is so until the small volume A"_, B;;_, A'B' which exerts at point M an action F" = £pA"_iA'âfaj. All these actions having approximately the same direction, their resultant $ is equal to their sum and we have 4> = £p(AAi-i- Al A2 -f- . . . -T- A"_i A')\ Hi àç ) ^ d^\ H^ dw) " ''' (') Lamé, Sur l'équilibre des températures dans un ellipsoïde à trois axes inégaux {Journal de Liouville, t. TV, p. 126; 1889 ). CIIAP. VII. - TRANSFORMATION OF THE EQUATION AV = G. 'iOI Let's put The equalities (i5) will then give us H, H, I s/M7i) - {v-w), {w - m), and the equality (lo), transformed from AV = o into elliptical coordinates, becomes -(..-.)v/=:iH:(^^[v/=Â(7)f] H ~ ^- v/- .ïi.t.v^'^^l.v'*'/' Ha H I '/- ^(p) Hi * V^ù>i.(tvyè)l(^a) HHi H, 2 ./ - 'R ( I, K'rt / ,. \ (3o) Let V a function of the sevde variable u : t? a function of the sevde variable ç; ^ a function of the single variable iv, these three functions being determined respectively by the second order differential equations (3i) W dw where A and B are two arbitrary constants. Let us then pose (32) Y = V)'<>^ and substitute this expression in equality (3o); this one of a02 MVRK II - DISTRIDUTION AND PROBMiMK OF DIRICIILET. will come t)\!P'C^[(A + BiA)(p-"0-4-CA + Bp)((P - f/)-f-(.V + Bw)rfi - t')] = o. She will be equally satisfied. Now, Lamé has shown that the equations (3i) can be integrated by certain series developments whose coefficients can also be determined in such a way that the function V, obtained by the combination (32), reduces to Q for u =^ c^. The Dirichlet problem is thus solved for the space outside the ellipsoid. The fruitful path that had led Lamé to this beautiful result, namely the transformation of the equation AY = o into orthogonal coordinates, allowed Lamé and other analysts to solve Dirichlet's problem for a large number of cases, and in particular for any space, bounded or unbounded, that confines only to a surface of the second order (' ). (' ) See Heine, Handbuch der Kugelfunktionen, -i" edition; Berlin, 1878 niAP. vni. rA METHODE DK L IXVEKSION. -20'i CHAPTER VIII. THE METHOD OF INVERSION. v5 1 - The method of inversion or electrical images. We know the immense role played, in modern geometry, by 1(!S methods of transformation which make it possible, when a problem is solved for a certain figure, to deduce other figures for which one can immediately solve the same or similar problems. Sir \V . Thomson (' ) has shown, and this is certainly one of the beautiful discoveries of this great geometer, that when one knows how to solve the Dirichlet problem for a certain space, one can form an infinite number of other spaces for which one can also immediately solve the Dirichlet problem. The method of transformation which he used under the name of the method of electric images is also known as the transformation by inversion or the transformation by reciprocal vector rays. Let .r, j', z be the coordinates of a point M in a certain space Fis. 37. and p its distance to Torigine O ifig' 37). Let's match it (') Extract from a letter from Mr. William Thomson to Mr. Liouville {^Journal de Liouville, t. X, p. 364; i9,!\b) - Extract from two letters addressed to Mr. Liou\?ille by Mr. W. Thomson (Journal de Liouville, t. XII, p. 256; 1847) - Liou 2o4 LIVIŒ II. - DISTRIBUTION AND DIRICHLET PROBLEM. a point M'{x',y\z) of a second space by the following formulas (,) a^^psf:., y^p^r^, --p-y^' A" being a positive or negative coefficient. Let p' be the distance OM'. It is easy to deduce from the formulas (i) that the product pp' is equal in absolute value to k, and that the point M' lies on the line OM, on the same side of the point O as the point M if A" is positive, and on the opposite side of the point O if k is negative. The transformation in question is reciprocal, in that we can write, instead of the formulas (i), the formulas (i bis) a7'=p'2|, y=p'2^;, ^'=P''|' Some of the simplest properties of this transformation are studied in the elementary courses. It is known that a plane transforms into a sphere in general, and into a plane in the particular case where the pole of transformation is on the plane; a sphere transforms in general into another sphere, which is concentric to the given sphere when the pole of transformation is the center of the given sphere, and which, when the pole is taken on the given sphere, reduces to the plane tangent to the given sphere at that point. This transformation has, from a geometrical point of view, a fundamental property that we will first highlight. Let MMi be a linear element of the first space, joining the point M(a;,y, z) to the point Mi (x 4- dx, y -+- dy^ z -^ dz). It will be transformed into a linear element M'M'j of the second space, joining the point M' {x',y, z') to the point M', (x'-h dx' ,y' 4- dy' , z'-\- dz') . According to the equalities (i), we have -, , / dx 9. X due \ dx = A - ^ , Vp2 p3 ; p2 p^ , , , l' dz "xz da dz ^ k { - - - - T p2 p3 VILLE, Note on the previous article {ibid., p. 265). - All these articles are reproduced in W. Thomson, Reprint of papers on Electrostatics and Magnetism, Art. XIV cil VI". VIII. L.V METHOD OF iNVERSIOX. 2o5 which gives A'2 r dx"^-^dy'-^-\-dz"^= ~ dx'^-h dy'^-t- dz'^ P L - ^{x dx -i-y dy -*■- z dz) - 52)(^p)2J r But we have X dx -^- y dy -^ zdz - Ç)d^, so that the previous equality reduces to ^ ■!. ) dx">- -f- dy"^ -I- dz!^ - -- ( f/j^s _i_ dyi- -f- c?^^ ) P or 2 lï 2 M'M'i = _, MMi . OR Let's see the consequences of this equality. From point M, let us radiate an infinity of linear elements MMi, MMo, MM;),.... They will have as correspondents an infinity of linear elements M'M', , M'MJ,, M'M!, all coming from the point M', and we have M' m; _ M' M', _ M' m; _ _ \k\ "MM7 ~ "MmJ ~ "MM^ - - - - - =^2 - The infinitely small figure formed by the points M,, M2, M3, . . . will transform into a similar infinitely small figure formed by the points M',, M.,, M'^, .... We express this property by saying that the space (^', 7'', z') (which the inversion transformation makes correspond to the space [X, y, z) is a conformal representation of it. 11 It is easy to see that this result leads to the following ones: Two curves or two surfaces that intersect at point M at an angle a are transformed by inversion into two curves or two surfaces that intersect at the same angle a at point M' transformed from point M. In particular, three families of orthogonal surfaces transform into three other families of orthogonal surfaces. Uo6 I.IVRK 1 . - DISTRIBUTION AND DIRICIILET PROBLEM. Let be a point M' of the transformed space, and let u^ v, w be its cartesian coordinates. The position of the point M which corresponds to it in the first space is fixed by the knowledge of these three quantities u, v^ w. We can therefore consider these three quantities as three curvilinear coordinates fixing the position of a point in the first space. The point M' is at the intersection of the three orthogonal planes The point M must therefore be at the meeting of three orthogonal spheres. This is easily verified, because the three previous planes give, by transforming themselves, according to the equalities (i his)^ the three surfaces '^ V , , ^^ x^- -+- Y- -^ [ z^ - = O, \ 2(1'/ 4 H' 2 which are three spheres intersecting at point O and tangent respectively at this point to the planes ZOY, XOZ, YOX. The coordinates m, c, w constitute, for the point M, a system of orthogonal curvilinear coordinates. The preceding equations provide the transformation formulas that allow us to go from Cartesian coordinates {x^y^ z) to curvilinear coordinates (m, p, w) or vice versa. These formulas can be written again (3) i X'- ^ y^- -\- z'- - k - a: = 0, II ' a:---HjK-+2' - x-^y'^+ z"' - \ ^ k - z -0, w Equality (2) gives us But, if we put du^ -+- di>- -\- dw^ = j- ( dx^ -h dy''- -\- dz-). p'2="2-1- {>^--\- w'i CHAP. MU. - I.A .MKTHODE DK L INVERSION. 'loy we know that p2p'2= A:2, nd the previous equalizer should be (2a) dx'^ H- dy'>- -\- dz^= -^ ( du^ -+- dv^ -\- dw^- ). If we refer to equality (8) of the previous chapter, we see (|that we have, for the change of coordinates that we consider, (4) H2= H?-=r H|= 4J Now, let us consider a function \ (.r,jKj ^) which is harmonic inside a certain surface S belonging to the first space; for all values of x^y^ z, which correspond to the points inside the surface S, we have W(x, r, z ) - o. ^the points {x,y, z-) inside the surface S correspond to a set of values of the quantities u, v, w. Let ^{ii, p, a) be what the function V becomes when we replace a:, y, z by their expressions in terms of u, v, w. We will have, according to equations (i bis), For the set of values of u, v, (v, that we have defined, this function will verify the transformed equation of AV = o, that is, according to equations (4), ) / I of I> > du \p'2 du / to / I d<\> l ^ '^ , / i ')1> 1 -t- y- 1 \p'- div This can still be written p' Idu^ Vp' / "^ dv^ \'f') '"' à^^\ô' 2Ï El 1 ^ ^ 1 - ^^ 1 p' [^àu^ p' ' dv^ p' ~^ àiv^ p' If we put ai d' d^ 208 BOOK II. - DISTHIBUTION AND PROBLEM OF DIRICIILET. and if we notice that A' " A - = o, P this equation will simply become (6) ^'(?)=" The result we have just obtained is susceptible to an interpretation that makes it important. The coordinates u^ v^ w can be considered as the Cartesian coordinates of a point M' in a space transformed by reciprocal vector rays of the space to which the point M(^,j^, ^) belongs. This transformation makes the surface S correspond to a surface S'; any point inside the surface S to a point outside the surface S'^ to the points infinitely close to the origin in the first space the points infinitely far from the origin in the second space. If the function Y(.r,jK> ^) is harmonic inside the surface S, the function iy) Wiu, V, w) = -, *(", P, w) = -, V i-^^, ^, -,-^j will be, according to equalities (6) and (5), harmonic outside the surface S'. If the function \(x,y,z) takes, at point O, the value Vo, the function W(", ç, w) will tend to zero as ~ when p' grows beyond any limit. If the function Y[x,y,z) takes the value A at the point P of the surface S, the function W (m, t^, w) will take at the point P', transformed from P, on the surface S', the value OP' OP, These results easily lead to the following proposition: If we know how to solve the Dirichlet problem for V space inside a closed surface S, we know how to solve it for the space outside the surf ace S' transformed from S by reciprocal vector rays. C.llVl'. Mil. - l,.V METHOD OF INVERSION. 209 Let Q be the function, defined at each point of S', to which the sought harmonic function must become equal at any point of S'. Let us form, which we know how to do by hypothesis, a function V(x,y,z), harmonic inside the surface S, and taking at any point P of the surface S the value OP Then let's form the function Wiu,i^,w)=^,\('^^, ^.^ I -. /Ali kv kw P Vp'' p The function W(", v^ w) will be harmonic at any point outside the surface S'; it will become equal to O(P') at any point P' of the surface S'; it will behave at infinity as a potential function; it will thus solve the Dirichlet problem for the space outside the surface S'. It would be shown in the same way that, if Von knows how to solve the Dlrichlet problem for V space outside a surface S, we know how to solve it for the space inside the surface S', transformed from the surface S by reciprocal vector rays. Some consequences of this important proposal: We have seen that the Dirichlet problem can be solved for the space inside a spherical surface and also for the space outside this same surface; by what we have just demonstrated, each of these two solutions can be deduced from the other. We know how to find the Green's function for the inner space of a dihedron commensurable with u; we know how to solve the Dirichlet problem for the outer space of two spheres which intersect at an angle commensurable with tt. We know how to find Green's function for the space between two parallel planes; we can therefore determine the electrical distribution on two equal or unequal spheres that touch each other. The determination of the electrical distribution on two touching spheres is indeed one of the beautiful results to which is D. - I. i4 DISTIUBUTION AND PROBLEM OF DiniCllLET reached Poisson (' ). A great number of geometers, among whom one can quote especially Plana, Sir W. Thomson, G. Kirchlioff, have subsequently dealt with this problem. A remarkable exposition of the solution can be found in the work devoted by É. Mathieu to the exposition of the Theory of Potential (-). § 2 - Application. - Electrical distribution on a spherical cap. One of the beautiful applications that Sir W. Thomson (^) has made of the method of inversion is the determination of the electrical distribution on an infinitely thin conductor having the shape of a spherical cap. We shall state the solution of this problem in the form given to it by Lipschitz ('■), confining ourselves to the case where the spherical cap is removed from all influence. The solution we are going to develop is based on a few remarks that we will indicate first. Let S be a spherical surface having its center at the origin of the coordinates and a radius equal to R. Let W be a function which behaves at infinity as a potential function, which is harmonic inside the sphere, harmonic outside the sphere, but discontinuous on the sphere, so that in the vicinity of a point M on the surface of the sphere it takes a value close to W/(M) inside the sphere and a value close to We(M) outside the sphere. Such a function is not the potential function of a certain (' ) Poisson, Mémoire sur la distribution de l'électricité à la sur/ace des corps conducteurs {Mémoires des Savants étrangers, p. 1811, p. i). (^) E. Mathieu, Théorie du potentiel et ses applications à l'Électrostatique et au Magnétisme, t. II, p. 65; Paris, 188G. (') W. Thomson, Extracts from two letters to M. Liouville {Liouville's Diary, vol. XII, p. 243; 18^7. - W. Thomson, Reprint of papers on Electrostatics and Magnetism, 1" edition, p. 162 ). W. Thomson, Determination of the distribution of Electricity on a circulçir segment of plane or spherical conducting surface, under any given influence ( W. Thomson, Reprint of papers on Electrostatics and Magnetism, 2" edition, p. 178). (*) Lipschitz, Uebkr die Vertheilung der stâtischen Electricitàt in einem kreisformig begrenzten Segment einer Kugelflàche {Rorchardt's Journal, JBd. LVIIl, p. i52; 1861). CII.VP. VIII. - L.V METIIODK OF INVERSION. 211 tle of electricity distributed on the spherical cap. Let's transform the space by inversion by taking the pole of inversion at the center of the sphere and the square R^ of the radius for the coefficient of inversion; any point M of the sphere will be transformed into itself; any point inside the sphere will be transformed into an external point and vice versa. Consider the function w'(^',y,.') = ^w(î^ '7T R2 It will be harmonic inside the S sphere, harmonic outside the S sphere, but discontinuous on the S sphere. In the vicinity of the point M of the sphere S, it will take, inside of the sphere a value close to ^ We(M), and outside the sphere a value close to ^ W,(M). Consider then the function = W{x,y,z) It will be harmonic inside the S-sphere, harmonic outside the S-sphere, and continuous on the S-sphere, so that this function will be the potential function of a certain amount of electricity distributed on the S-sphere. An analogous method would apply to the more general case below: A closed surface is composed in part by a portion S Fig. 38. RW'ix,y,z) R",/R2- R2.r R2g P' (the spherical surface, partly by another surface S of shape* ([uelconcic (/?^. 38). The function ^/(x, y, z) is harmonic both inside and outside this closed surface; it is ai-i DISTRIBUTION AND PROBLEM OF UIRIOIILET. behaves at infinity as a potential function; it varies in a continuous way when passing the surface S and in a discontinuous way when passing the surface S. Let O be the center of the sphere S taken as the origin of the coordinates; let R be its radius. The function (8) W(^,7,.') + -W( - ,^,- V(^, j, -s) will be the potential function of a certain amount of electricity distributed on the closed surface composed by the surface S, the surface S and the surface S' transformed from the surface S by reciprocal vector rays. It is a process of this kind that will allow us to determine the potential function of electricity in equilibrium on a spherical cap. Let S be this spherical cap {^/îg- Sg); let R be its radius; soil H is the distance of the base circle S from the center O of the sphere. The radius of the base circle is the quantity (9)" = (R2-H2)i Consider the quantity u defined by F equation (lO) (z-BY^ "2-1- u and consider the function (it) V(a:-,7, s) = K( ^ - ai-ctang^ where K is any constant CHAP. VIII. - THE METHOD OF INVERSION. "ilS According to what we have seen in Chapter VII, § 3, this function is harmonic in all space, except on the base circle (the cap; it behaves at infinity as a potential function; it varies in a continuous way when we cross this circle, at a finite distance from its edges 5 on this circle, it takes the value -;- - When crossing the circle, the partial derivatives of the first order of this function change sign without changing absolute value. Let T be the closed surface formed by the cap S and the plane S. Let us take a function 'W[x,y,z) equal, inside the surface T, to and, outside the surface T, at bow tane a / K \^ ^ - arc tang ^ This function will be harmonic inside the surface ï ; it will be harmonic outside the surface T ; it will vary in a continuous way when crossing the surface S and in a discontinuous way when crossing the surface S. Its first order derivatives will not undergo any discontinuity when crossing the surface S. Let's consider the function If we define the quantity u' by the relation (10 bis) l. u z H = P*("'^-f-i<') \ I p'^ / R It is easy to see that we will have, inside the surface T, K "2 KR 1 ' Tt u arc tang - "a y /H - - arc tang - P K-^ " a (--2) \{x,yz) inside the volume between the surface S and the cap 2'. -il 4 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. into which it is transformed by inversion, / - \ ' - \ , . ",, ,A T. u- \ KR / TT a'- 1 [vihis^ V(a:,jK,z) = K l - - arc tang - / -; - \- arc tang - ^ and, outside the surface T', formed by the caps S and S', s Tr, ■ ^ri"^ u-\ KR Tz * u'"' ( 12 ter) y (x. )-, ^ ) = K \ arc tang - / -r- - \ - f- arc tang - ^ ■ \ 2 a; p \ 2 o, This function will be, according to the above, the potential function of a certain amount of electricity distributed on the surface S, on the surface S and on the cap S', into which the plane S is transformed by inversion. But the first-order partial derivatives of this function vary in a continuous way when one crosses either the surface S, or the surface S'; the electric density is thus null on each of the two surfaces S, S', and the preceding function is the potential function of a certain quantity of electricity distributed only on the cap S. Moreover, on this cap, \ ix^y^ z) takes the constant value Ktt. The distribution in question is therefore an equilibrium distribution. To complete the determination of the potential function of a quantity Q of electricity distributed in equilibrium on the cap S, it is sufficient to establish the relation which connects the constant K to the total load Q. To obtain this relation, we will write the value of the potential function at the point O in two ways. This value can be written, on the one hand. On the other hand, it can be deduced from the formula (12). Equality (10) shows that at point O, we have u = H2. Equality (10a) shows that at the same point O we have , R2 CIIAP. VIII. - THE METHOD OF INVERSION. '215 In fact, we have i it "'" a arc tang - = arc tang - - > 2 a ,\ and, in the current calculation, I / TT u '■ \ a - \ arctang - / "^ tt' p \ 2 a J H So we have Vo = K ( - -h arc tang _ ) -h K cr . \'i. a j K Let y be the angle under which the radius of the base circle is seen from the center of the cap; the equalities a . a -^^sinY, jj = tangY allow to write Vo = K(7: - Y -+-sinY). Let's compare the two values of Vo, and we finally find Q (i3) K R(Tr - Y "T~ sinY) This equality completes the determination of the potential function of a quantity Q of electricity spread in equilibrium on the spherical cap S. This amount of electricity raises the potential level of the spherical cap to a value Kt:, or, according to equality (i3), to a value Q 7t R T. - Y"^ sinY The capacity of the spherical cap is therefore (i4) C = R î ". § 3 - Liouville's and M. P. Painlevé's theorems. One can, in an infinite number of ways, obtain a conformal representation of a plane (^, y) on another plane (X, Y), that is 2l6 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. find transformation formulas ^=?(X,Y), such that if the two curves represented with respect to the axes of ce and j" by the equations intersect at angle a, the two transformed curves, represented, with respect to the X and Y axes, by the equations F(X,Y) = /[c?(X,Y), .KX,Y)l = o, G(X,Y) = ^[(X,Y,ZK x(X,Y,Z),j is regular and satisfies the equation of ^W a^W (92 W _ at any point in the domain D, transformed from d into V space rfe,v(X, Y,Z). In answer to this question, we can say that the transformation formulas we are looking for necessarily represent a transformation by reciprocal vectors followed by a change of axes of rectangular coordinates. This important theorem, by clearly marking the exclusive role of the method of inversion, narrowly limits the field of Analysis in the way opened by Sir W. Thomson. It is due to M. Paul Painlevé (' ). (') Paul Painlevé, Sur la transformation des fonctions {x, y, z) qui satisfont à l'équation AV = o et sur les coordonnées curvilignes orthogonales ( Travaux et Mémoires des Facultés de Lille, l. I, n° I ; 1889). DISTRIBUTION AND DIRICHLET'S PRONLEM. CHAPTER IX. THE METHOD OF MR. CARL NEUMANN. § 1 - The theorem of Mr. Vito Volterra. Lejeune-Dirichlet's demonstration does not allow us to affirm with complete certainty that the problem to which this geometer's name has been given admits, in all possible cases, a solution. This demonstration only has the effect of replacing the hypothesis that this problem admits a solution by other hypotheses which are perhaps more probable, although also not very well demonstrated. Green posed the same problem in a different form, but without advancing the solution. Thus, far from having a general method to solve the Dirichlet problem, we do not even know how to demonstrate the general existence of a solution of this problem. It is therefore necessary to try to solve it in particular cases. There are a very small number of special cases for which the Dirichlet problem can be solved immediately (Chap. VI). The use of appropriate orthogonal curvilinear coordinates has enabled Lamé and other geometers to extend considerably the number of cases in which it is possible to solve Dirichlet's problem. These methods, however, have so far proved fruitful only for surfaces that are part of an orthogonal and isothermal curvilinear coordinate system (Chap. VII). The method of inversion, created by Sir W. Thomson, has greatly increased the number of cases in which the Dirichlet problem can be solved; but the power of this method cannot be shared by any other method of transformation, as M. P. Painlevé has shown (Chap. VIII). CHAP. IX. - THE METHOD OF MR. CAHL NEUMAXN. '121 The efforts we have just traced only allow us to solve the Dirichlet problem for particular classes of surfaces, very large classes, it is true. The methods we are now going to explain are profoundly different from the previous ones, in that the bodies to which they apply are not required to belong to certain classes, but only to verify certain restrictions. They are therefore infinitely more general in scope than those we have explained so far. They have, it is true, a disadvantage over the previous ones. To solve the Lejeunc-Dirichlet problem for the sphere, for the ellipsoid, etc., the methods we have indicated lead either to integrate over the whole surface an analytical combination in which the given function appears on this surface; or to determine, by means of the known values, that the function sought takes on the surface the coefficients of a certain development in series. The actual use of these methods gives rise to arduous, but nevertheless practicable, calculations. Thus, the development in series of spherical functions which are used to solve the Dirichlet problem for the space inside or outside a sphere give rise to numerical calculations in celestial mechanics and in the study of terrestrial magnetism. It is different for the methods that we are going to expose. These methods lead to the representation of the sought function by a series of integrals extended to the given surface. Only the first of these integrals depends directly on the values given on the surface. Each of the other integrals requires, in order to be performed, that we know the one that precedes it. Such a procedure undoubtedly escapes any numerical calculation. Therefore, it loses all practical interest; but it is nevertheless of great theoretical interest, because it demonstrates, without any possible dispute, the existence of a solution of the Dirichlet problem for the extended classes of surfaces to which it applies. A method, invented in i833 by Muiphy (') to treat the problems of electrical influence, gave the first idea of ( ' ) MuRPHY, Elementary principles of the theories of clectricity, heai, and inolecular actions^ Pari, l, Cliap. V, p. gS; i833. ■i-Vl I.IVIŒ 11. - DISTUlHliTION lîT PnOBLbME Olî UlRICIILET. this class of processes. This idea was further developed by Béer (') in i856. But it is to M. CarlNeumann (-) that we owe the legitimacy of such processes. The convergence of the series employed can always be demonstrated by means of a fundamental theorem which, implicitly contained in the work of M. Schwarz and M. Cari Neumann, seems to have been stated for the first time by M. Vito Volterra (-'). Mr. Axel Harnack ('') and Mr. Paul Painlevé (^) have insisted on the role that this theorem must play in the study of harmonic functions of two or three variables. Here is the statement of this theorem: Let Vt, Uo, ---, U", ... be an unlimited sequence of harmonic functions inside a certain space E bounded by a surface S. Let Ut, u^-, --., Un, ... be the values towards which these functions tend when the point at which they are brought tends towards the surface S. If the series (l) Ui-\- ll^-T-. . .^ U,i-\- ■ - ■ converges uniformly at any point on the surface S, the series converges uniformly at any point in V space E and represents^ at any point in this space, a harmonic function. Let us first prove that the series (2) converges uniformly in the space E. (') BTÊ.Y.B., Allgemeine Méthode zur Bestinimun g cler elektrischen und inagnetischen Induction {Poggendoff's Annalen, B. XCVIII, p. 187; 1806). (^) C. Neumann, Untersuchungen ûber das logarithmische und Newton'sche Potential (Leipzig, 1876). (') Vito Volterra, Sopra alcune condizioni caratteristiche délie funzioni di una variabile complessa {Annali di Matematica, series II, t. XI, p. Sa: 1882). (' ) Axel Harnack, Die Grundlagen der Theorie des logarithmischen Potentiales un der eindeutigen Potentialfunktion in der Ebene, § 20; Leipzig, 1887, {') Paul Painlevé, Sur les lignes singulières des fonctions analytiques (Annales de la Faculté des Sciences de Toulouse, l. II, p. B. i5; 1887). ClIAP. IX. - THE METHOD OF MR. C.VRL NEUMANN. 223 Let's consider the function *"/. = U" -T- U,i+i -h . . . + U "+p. It is a matter of proving that we can always take n large enough so that we have, whatever p is, l*"pl<^, î being any positive number given in advance. Now, the series (ij being uniformly convergent, if we pose we can always take n large enough to have, whatever/? On the other hand, the quantity ^np, the sum of a finite number of harmonic functions in E space, is a harmonic function in E space. The values it takes in E space are, according to the general theorem of M. Cari Neumann (Book II, Chap. III), between two of the values it takes on the surface S. On the surface S it becomes identical to cp . If therefore, at any point on the surface S, we have I ? "p I < ', whatever/?, we will have, at any point of the spaceE, the ^npKh whatever/?. So let it be V = Ui-+-U2H-...+ U"^... the continuous function whose existence is now proven. We have to prove that this function is harmonic at any point inside the space E. Let M be a point inside the space E. Let us take it inside a whole sphere S contained in E. Let G(M) be the Green's function for the point M and the sphere S, a function which is known (see Chap. VI). Let's put w" = u, + u, + ...+ u", V = W"-f-R". 224 BOOK II. - DISTRIBUTION AND DIRICLLET PROBLEM. The function W" being obviously harmonic inside the sphere S, its value at any point M inside the sphere is given by This equality can be written The first member does not depend on n; on the other hand, we can take n large enough so that R", and consequently the second member, are as close to O as we want. The previous equality can therefore only exist if we have ^"-<-)=S.^-'-^'''^ The function V is, therefore, a harmonic function at any point inside the sphere S. The function V is harmonic inside any sphere contained in the space E; that is to say that it is harmonic at any point of the space E. Our theorem is thus proved. Let us add to this theorem the following proposition: When the point M of V space E tends 'towards the point m of the surface S, the value V(M) that the series at point M tends to the value v[ni) that the series Ml -+- Mo -I- . -1- M" -t- . at point m, and it reaches this value when point M comes to point m. To demonstrate this, let's say W',, - "1 -i-"2 -^ - - .+"rt, We can take 'e n large enough to have, for THE METHOD OF MR. CARL NEUMANN. Any position of the point M in the space E, and for any position (the point m on the surface S, any neighbor of o that is e. Moreover, when the point M tends to the point m, U(, Uo, \},i tending respectively towards w, w., ---, w", W" tends towards iv" . We can therefore, around the point m, draw in the space E a domain small enough so that, for any point M of this domain, we have |W"(M)-"'"(m)| -^ - - > - ^ - , - j-^ - > - - - are obtained by multiplying V(P) by factors which are finite for any point M in the space E. Having proved this lemma, let's consider functions in numbers unlimited U,(M), U2(M), UaCM), .... Suppose that each of these functions is harmonic within V space E and that each of them has, in all this space, a constant sign which is the same for all. If the series Ui(P) + U2(P)-i-U3(P) + ... is convergent at any point P of V space E, the series Ui(M)-4-U2(M)-+-U3(M)-i-... converges uniformly at any point in this space and represents, in this space, a harmonic function. According to the previous proposition, we have U,(M) = X,U,(P), U2(M) = X2U2(P), dUi(M) = [^iU,(P), ) = viUi(P), t)U2(M) d.T = M.2U2(P), * - ? = V,U2(P), \^, lo, ..., jx,, [Xo, ..., V,, V. point M of the space E. Then we have . being finite factors for all = X "U"(P)+X "+iU "+i(P)-.-. dx ôx dx .a "U"(P)+[Ji "+iU "+,(P) X "+/>U;i+p(P), ^"+pU "+;,(P), d2U"(M) a2U"^,(M) ()2U "+,(M) dx^ dx* " dx^ = v "U"(P)4- v"^,U"^-,(P) ^hi+p^ ii+i>{^ )i 228 BOOK II. - DISTRinUTION AND PROBLEM OF DIRICIILin. But the functions U,j(P), Un+i{^)j - - -, U/,^^(P) all have the same sign. If therefore 4^ is a finite factor between the largest and smallest value of X^, \i^i, - - -, "^n+p '■, if likewise OÏL is a finite factor between the largest and smallest values of [x", p."^,, . . . , [J-^+y?; if finally SIL is a finite factor between the largest and smallest values of v", ^n+\ ! n+p we will have U"(M)-hU "+i(M) dx dx ■U"^/,(M) ^U "+;,(M) ÔT -C, [U.(P)= 01L[U"(P) -U "4-i(P) + . U "+, (?)+.. .+ U"^"(P)1. d2U "m) . a2U "+i(M) dx'^ a^2 .+ ^^^^^^f^^=<î>C,[U"(P)-+-U "+,(P) + ...^U "+p(P)| Once these results are obtained, the proof of the stated theorem is completed without any difficulty. § 3 - Some definitions. If we cut a closed surface by any straight line, we can obtain as intersection points isolated from each other and also whole line segments. For example, if we consider the surface generated by the rotation of the line FABCD {fig. \o) whose side AF is perpendicular to DF, a line Any straight line will generally cut it into two or four isolated points; to the straight line AI will correspond as an intersection a segment; to the straight line AB, a segment AB and a point K, etc. Whenever the surface is cut by a line and one or more segments are obtained, it is advisable to count only the points of intersection as the points of intersection. CHAP. IX. - THE METHOD OF MR. CABL NEUMANN. 229 According to this convention, the line AI will cut the surface at only two points A and I; the line AB at three points A, B, K. .... In what follows, we shall constantly assume that a closed surface is intersected by any line in space at a finite number of points, the maximum of which is necessarily even, and we shall call this maximum number the rank of the surface. The surfaces that we will specially study are the second rank surfaces. We will admit that, except at certain isolated points or along certain isolated lines, the surface under study admits a tangent plane that varies from one point to another in a continuous manner. It is easy to see that, if the surface is of second rank, the tangent plane at each point will leave the whole surface on one side. At the isolated points where there is no tangent plane, we will admit that there is a conical tangent sheet; if along a line there is no tangent plane, we will admit that at each point of this line there is a tangent dihedral. We will say that a surface, open or closed, is multi-layered of order n when it will be possible to imagine in space n fixed points such that any plane lying on the surface passes through one of them, but when at the same time it will not be possible to assign less than n fixed points enjoying this property. A uniwoven surface is necessarily composed of a portion of a cone and, therefore, is necessarily open. Examples of closed bi-structured surfaces are the parallelepiped, the octahedron, the tetrahedron, the surface generated by the revolution of a rhombus around one of its diagonals, etc. In this last surface, the two stars, i.e. the two fixed points of the definition, are the ends of the diagonal considered; in the parallelepiped and the octahedron, the ends of any diagonal; in the tetrahedron, any vertex and any point of the opposite face. Non-biased second rank surfaces have a property that we will demonstrate and that will be useful later. 23o BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. Let us take a point [jt. on a similar surface; let m be another variable point of this surface; let dS be an element surrounding the point m; let N^ be the normal to the element û?S toward the interior of the surface S; let p be the line m [ji. The quantity n COs(p, Ne) ^g has a value given by Gauss's third lemma; this value is air if the surface S admits at the point [x a continuously variable tangent plane; if the tangents at this point form a conical sheet of spherical opening a, it has the value a; if, at the point u, it admits a tangent dihedral whose opening is -j it still has the value a. We can therefore write, in all the possible cases. S If we observe that, for the considered surfaces, cos(p,]N, is never negative, we see that we will also have S '^'^^^p'^^^^.s.^, integration extending only to a portion o- of the surface S. If p. and [jl' are any two points, distinct or coincident, on the surface S, and if the surface is divided into two portions ' inequalities that certainly give Scos (p,Ne) , n cos(p', N,)^^,^ Ar p2 "^ O p'2 - L2 This inequality is established by assuming that the two points [x and a' have a given position on the surface S. But the quantity L has a value independent of the position of the two points jx and [x'. It is obvious that we can always make sure that the two quantities A and S" do not fall below a certain limit. It will thus be possible, whatever the positions of the two points [x and jx' and the two segments o-, ([i)cos(r, N^) S. '^ '^ 1^ 3 ' On the other hand, the quantity S. [v(m) - v(ix)]cos(r, N^) d^ obviously varies continuously, even when the point M crosses the surface "t. We can therefore, around the point [jl, draw a domain d, contained in D, small enough so that, for any point Mj of this domain, we have .)] =- ap([ji;. If the two points M/, M^ tend towards the same point of the surface S, one remaining inside this surface, the other remaining outside it, we have, in all cases, (5) lim[U(M,)-U(Me)]-4Trp([x). § 5 Definition and properties of subordinate functions. Let '^{m) be a quantity defined as follows: If at the point m the surface s admits a unique tangent plane that is continuously variable, we have '^{m) ^^ o. If, on the other hand, the point m is a singular point, we have àÇm) = 2tz - a, a having the same meaning as in formulas (3a) and (4a). Let's put (6) ZTzv'(m) ^ u{jn) -^'b{m) -h v{m) and i>'(m) cos( r, N^) (7) U'(M}=g rî dS. This equality defines a function which exists at any point in space, and which we will call the subordinate function of the function U(M). It is easy to see from the inequalities (3) and (3 bis) that we have 2Trf'(/n) - lim[U(M,-)] - 7.Tzv(m). The subordinate function is thus a continuously variable quantity on the surface S. Also the function U'(JVl), defined by equality (7), is a function of any analogous point 236 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. to the function U(M) defined by equality (2). It is harmonic in all Fespace, except on the surface S. It is discontinuous on this surface; if w'(a) denotes the value it takes at the point [a of this surface, we have (8) Just as we formed the subordinate function U'(M) from the function U(M), we can form the inner subordinate function U"(M) from the function U'(M), We will then have a regular way to form an unlimited sequence of functions as shown in the following Tableavi: U (M)=§ v(tn)cos(r\ Ng) r2 ^S, iT.v" (m) - u' (m)-'\i(77i)v' (m), 2Tzv"'(m) - u"{m)-h'\i)o" (m), It is easy to see, by the equalities (3 bis), (8) and the analogous equalities, that we can write 1' lim[U (M,-)] =2t:[p ([jl)-^p' ([x)J, Iim[U'(M,)]-27i[p'(»-4-(."(fx)]. Mim[U"(MO] - ■27r[p"(!x) + p"'([Ji)], On the other hand, according to equalities (4a), (8) and similar equalities, we can write ( liin[U (Me)] = 27r[p'([a)-P (fji)], lim[U'(M,)]==27r[p"(jx)-(>'(|x)], lim[U"(M,)]-27r[p"'([Ji)-(^"([x)], (IIJ Let us now turn to the fundamental proposition of this whole theory; it is stated as follows: If G is the largest and H the smallest of the values that CUAP. IX. - THE METHOD OF MR. CARL NEUMANN. 237 takes v(ni) on the surface S; if u. and jjt,' are any two points on the surface S, we always have i.'(}x')-i.'((x)^(G-H), which is the announced inequality. Let be, in particular. G' the largest and H' the smallest of the values that ç'(m) will take on the surface S; we will have iG'-U')<'k(G - U). If G, G', G", G"', . . . are the largest values and H, H', H", H'", . . . are the smallest values of the functions v{m), v'{m)^ t'"(m), v"'(m), ... on the surface S, we have (IV) (G'- H')f.X(G - H;, (G" - H")1X(G'-H'), (G'"- H"')"{m), . ..., U, U', L", ..., according to the rules indicated in table (I); we will look for the constant C, limit of the values of the quantities i^{m), v'{m), ç"{m), ...and the function we are looking for will be given by the series (VIII) j V(M,) = C+^|[U(M,)-U'(M,)] + [U" (M,,-IJ"'(M,)] The Dirichlet problem is thus solved for the space inside any non-biased second rank surface. § 7 - Solution of the Dirichlet problem for the space outside a non-biased second rank surface. The Dirichlet problem can also be solved for the space outside a non-biased second rank surface; but the solution presents some complications that were not available in the previous problem. Let w{m,) be the set of values given on the surface S. Let us place inside this surface, at an arbitrarily chosen point, an electric charge k. At the point m of the surface S, the potential function of this charge takes the value q(^m). Let's put \>{}n) =■ iv{ni) -\- q{}n), and form the function U and the subordinate functions as shown in table (I). The function v{m) is the sum of a function w(^ni), independent of A, and a function q{m'), proportional to k; it is therefore a linear function of k. It is easy to see that it is CH.VP. IX. - I..\ METHOD OF M. CARI. NEUMANN. 24 1 is the same for the functions v'{/n), v"{m), ..., so that the limit of these functions is L + PA", L and P being two constants. Let us now consider the series - [U(Me) ■+■ U'(M,) -h U"(M,) 4- U"'(M,) - . . . J. 27t Each of its terms is a harmonic function outside the surface S, behaving at infinity as a potential function, and, according to the equalities (III), when the point M^ tends to the point m of the surface S, each of these terms tends in a continuous way to the corresponding term of the series (lo). Consequently, the series considered represents a harmonic function in all the space outside the surface S, behaving at infinity as a potential function, taking on the surface S hi value v{m) - G = iv(m) -^ q{/n) - L - P k and continuously tending towards this value when the point M^ tends towards the point m. This result leads us without any difficulty to the following rule: If we want to form/' a harmonic function at V outside the surface S, behaving at V infinity as a potential function, continuous up to the surface S and taking at any point m of this surface a given value w(m), we will operate as follows: At a point O arbitrarily chosen inside the surface S, a charge K will be placed whose potential function will take the value q{ni) at point m. Posing v{m) - w{m) -r- g{>n), we will form the table (I). The functions v'{m), v"(m), ... will tend to a limit of the form L + PK, h and V being two constants. D. - I. i6 242 BOOK II. - DISTKIBUTION AND PROBLEM OF DIRICIILET. In the functions U, U', U", ..., we replace by the value - ^- If Von denotes by R the distance from the point M,. to the point O, and if we form the function (IX) V(M,)=^ - _L[U(M,)-f-U'(M,) + U"('Me)H-...J, r ri air it will solve the problem. We can thus see that the method of M. Cari Neuinann allows us, given a non-biased surface of second rank, to effectively solve the Dirichlet problem both for the space inside this surface and for the space outside it. It is not true that we have the solution of the Dirichlet problem for all possible cases; but, at least, we obtain the solution for a case infinitely more extensive than all those where it had been possible to give it until now. NATURAL DISTRIBUTION. 9.f'i CHAPTER X. NATURAL DISTRIBUTION. § 1 - How the solution of the Dirichlet problem for all bodies can be reduced to the search for the natural distribution over all bodies. If we know how to solve the Dirichlet problem for the space inside a certain surface, we also know how to solve it for the space outside the surface into which the first one is transformed by inversion. If, therefore, we know how to solve the interior Lejeune-Dirichlet problem for all bodies, we also know how to solve the exterior Lejeune-Dirichlet problem for all bodies. We can only propose to solve the first problem. We can solve the Lejeune-Dirichlet problem for the space inside a certain surface S, if we can find the Green's function relative to this surface S and to any pole P inside this surface S. Finding the Green's function for the surface S and the pole P is the same as finding a function TÇa^y, z), harmonic inside of the surface S and becoming equal to ( - -^=- j at any point M of the surface S. Let's transform the surface S by inversion by placing the transformation pole at the point P and taking the unit for the transformation coefficient. Let S' be the transform of the surface S. Suppose we know how to find the function -^{0:',^', z'), which is harmonic in the space outside the surface S', equal to unity on the surface S' and, at infinity, equal to zero like a potential function. If we transform the surface S' into the surface S again by inversion, the function f{x',y, z')is transformed into a function g{^,y, :■)■ If we denote by /the distance of the point a44 BOOK II. - DISTRIBUTION AND THE DHUCIILET PROBLEM. {■r,y, z) at point P, the function will be harmonic at any point inside the surface S and will take the value ( - -=- ) at any point of this surface. This will be the \ MP / function sought Y{x,y, z). One will therefore know how to find the function Tl^x^y^z) for the space inside any surface and for any position of the pole in this space, if one knows how to find, for any surface, the function Y(^, jK? z) which is harmonic in all the space outside this surface, which takes the value i on the surface, and becomes, at infinity, ('■gale to zero as a potential function. This function ^(^x,y,z) is the potential function of a certain quantity of electricity distributed in equilibrium on the conductor subtracted from any influence. The quantity of electincity which admits for potential function y(c,jKj ^) is the quantity C - j C being the capacitance of the capacitor and £ the constant of the laws of Coulomb. Let us suppose that we know how to determine the equilibrium distribution that alTecle, on the conductor subtracted from any influence, an electric charge equal to the unit. We can then calculate the potential function u{x, y, z) of this distribution. This function takes, at any point of the conductor surface, the same value A. We will have and ^=é G j((t, V. z) A We will name natural distribution on a conductor the (distribution that affects a charge equal to the unit to the surface of this conductor subtracted from any influence, and we can state this theorem: The Lej eune-Dirichlet problem can be solved for V space both inside and outside any surface, if one can determine the natural distribution on any conductor. CHAP. X. - THE NATURAL DISTUIBLTIOX. 1^5 § 2 - M. G. Robin's method for determining the natural distribution on a conductor. Any method suitable for solving the Lejeune-Dirichlet problem in general is obviously suitable for determining the natural distribution on a conductor in particular; but the search for the natural distribution gives rise to certain methods which cannot be extended to the solution of the Lejeune-Dirichlet problem in general. 13e this number is the elegant method indicated by M. G. Robin to determine the natural distribution on the surface of any body bounded by a surface of second rank. The starting point of M. G. Robin's research is the consideration of a remarkable functional equation which is satisfied by the electric density at any point of a conductor in equilibrium. Let t(M) the electric density at a point M on the surface S of an electrified conductor; N,. the normal to the surface S, at point M, to the outside of the conductor ; j, the component along N^ of the electrostatic action at point M. We know that we have (p. 88) "1>K,. = 27r£n/r 2 L7_ Plu" s PM 'T PM CHVP. X. - NATURAL DISTRIBUTION. '^49 But, according to equality (2). we must have /AIN C e(P)cos(PiVl, N^) ,ç 2re(M) = V - - ^ ab. PM If we put (6) e(P)cos(PiVl,Ne) , . ,.., ^ ^ -p- ^ - - di - 2Tree(M), Pi\r e(P)cos(PM,N") -) It is not assumed that we have given values to the function v{m) at the various points on the line L. We only assume that these values must be finite ('). It is easy to see that, under these conditions, the demonstration (' ) This restriction is essential. If the function v{in) is not subject to being finite along the lines of discontinuity, the proposed problem can admit an infinite number of solutions, as M. Painlevé has shown for the analogous problem relating to functions of two variables [Paul Painlevé, ^wr les lignes singulières des fonctions analytiques [Annales de la Faculté des Sciences de Toulouse, t. IL p. B.22; 1888)] CH.VP. XI. - THE ALTERNATING PROCESS. 253 by which we proved, in Chapter V, that the Dirichlet problem could not admit more than one solution remains entirely valid. But a circumstance arises to which it is necessary to call attention. Let us first take a point m in the neighborhood of which the values of v{m) do not experience any discontinuity. Let M/ be a point inside the surface S. Let V(M,) be a function which solves the Dirichlet problem for this space. We can always draw a domain D around the point m, inside the surface S, and small enough so that, for any point M/ of this domain, we have |V(M,-)-i^(/n)i<£, î being a ([uuntité positive given in advance. Let, in efi'et, s be the part of the surface S which forms the limit of the domain D (Jîg'. i'i) with a surface a- included inside do Fi the considered space. The function V(M) is continuous on the surface t and on the surface s. It is the same for the function U(M) = V(M)-t^("i), since v(/n) does not vary with the point M. We can therefore take this surface (-î + '^) small enough so that, for any point M on this surface, we have |U(M)|<£. Then, as the function U(M/), harmonic inside the domain D, takes its absolute maximum value at a point of the surface {s -t- a-) which limits the domain D, we see that we will have, as we had announced, |U(M,)|{m) and on the surface S4 the values m" This function takes, on the surface S3, values that we will denote by w™. (' ) Schwarz, Programm der Polyteknikum-Schule in Zurich; 1869-70. (") C. Neumann, Berichte iiber die Ver/iandlungen der K. Sàchsischen Gesellschaft der Wissenschaften zu Leipzig; 21 April and 3i October 1870. (') C. Neumann, Untersuchungen iiber das logarithmische und Newton'sche Potential. Leipzig, 1877. (*) Axel Haknack, Die Grundlagen der Theorie des logarithmischen Potentiales, .... Leipzig, 1887. D. - l. m 258 BOOK II. - DISTRIBUTION AND DIRICIILET PROBLEM. Let us form again a function l]3{x, y, z), harmonic in space (En- e), taking on the surface S, the given values p(m) and on the surface S3 the values u"^. This function takes, on the surface S4, values which we will designate by u\J. Continuing in this way, we will successively form the functions contained in the two tables below: I. - Harmonic functions in (E h- e). Functions. S,. Ui(^, jr, -s) i'(m) 1^3(^,7,-5) i'irn) Us(a?, jK, ^) i>(m) II. - Harmonic functions in (E'-+-e). S^ functions. Viix, j, z) ç(m) \],,(a!;,y, z) i>(m) l}6{oe,y,z) v(m) Values on S, S3. ..V It" "y u'I "y u "l We see that we must alternately form a function of the first Table, then a function of the second, which justifies the name of alternating process given by M. Schwarz to this method. We will see that the solution we are looking for can be obtained by taking, at any point M, the space E, (2) V(M/) = Ui(M,-) + [U3(M,)-Ui(M,)] + [U5(M,)-U3(M,- )]+...; at any point M/ in space E', (3) V(M,) = U2(M,0+ [U4(M,-)- U2(M,-)] + [UeCM/) - UiCM,-)] + - - - ; finally by taking arbitrarily, at any point of the space e, one or the other of these two expressions. Cn\P. XI. - THE ALTERNATING PROCESS. 9.Sçf § i. - Demonstration of the above theorem. To prove that the previous method provides the solution of the Lejeune-Dirichlet problem for the space (E + E'-t-e), it is obviously sufficient to prove the following propositions: 1° The series given by equality (2) is convergent and represents a harmonic function in the whole space (E + e). 2" The series given by equality (3) is convergent and represents a harmonic function in the whole space (E'h- e). 3" These two harmonic functions are identical in all of e-space. 4" When the point M/ tends to a point m of the surface S|, the function V(M/), defined by equality (2), tends uniformly to r(m). 5" When the point M,- tends to a point m of the surface Sj, the function V(M/), defined by equality (3), tends uniformly to v{m). The fact that the series given by equality (2) is convergent in the space (E -h e) and j represents a harmonic function can be deduced from the theorem of M. Axel Harnack, proved in § 2 of Chapter IX. The function U( (.r, y, z) takes on the surface S) the values p(/?i), which are all between G and K, and on the surface S3 the value K. The value of [^{x^y, z) at any point in the space (E + e) is therefore between G and K. This is especially true for the values denoted by u^^. The function \]2{x,y, z) has, in the whole space (E'+e), values between G and the smallest value of ('(m) or of m','. It is in particular the case of the values u!^ that it takes on the surface S3. It follows that the difference \\]Muiphy's problem kXdiTecheTchc of the combinatorial methods suitable for performing this reduction, because the first method of this kind, a method which later gave rise to the researches of Béer, of M. Cari Neumann, of M. Schwarz, on the Lejeune-Dirichlet problem, was given by Murphy (') in i833. § 2 - Fundamental laws of electrical condensation. Murphy's method was intended to study a special problem, the problem of electrical condensation, which can be stated as follows: Given a conductor C) {fig- 47 )> held at At the potential level Pi.', a conductor C2 connected to the ground and, consequently, maintained at the potential level O, find the distribution of V electricity on these two conductors. Before studying the method by which Murphj reduces this problem to the study of the distribution generated by given charges on each of the conductors put in communication with the ground, we will expose the fundamental properties of the distribution produced on the two conductors G| andCa- Several of (' ) Murphy, Elementary principles of the theories of electricity, heat and moleculars actions, Part. I, Chap. V, p. 98. 266 BOOK II. - DISTRIBUTION AND PROBLEM OF DIRICHLET. these properties will be invoked in the justification of Murphy's method. We already know (Chapter IV, theorem IX) that the distribution must be monogenous on each of the two conductors Cl and C2. On C, it has the sign of A and an opposite sign on C2. For the sake of clarity, we will assume that A is positive; the G| conductor will then be charged with positive electricity, and the C2 conductor with negative electricity. Lemma I. - If the conductor C2, put in communication with the ground, is subjected to the influence of a positive electric mass Y- concentrated in an external point M, it is covered with a monogenous and negative distribution. Let S2 {fig- 48) be the surface of the conductor Co. Let us surround the Fis. 48. point M of a small surface S. The potential function, equal to o on the surface Sa and at infinity, is positive and very large on the surface S. It is therefore positive in all the space outside the surfaces So and S. The quantity -7^ is then positive at any point of the surface S2, which proves the stated theorem: Lemma IL - If the conductor Co, placed in communication with the ground, is subjected to the action of the electric charges jji, [x', p.", . . . concentrated at the external points M, M', M", . . ., it is covered with a distribution which is the superposition of the distributions with which it would be covered, if it were subjected successively to the action of the only charge [x concentrated at the point M, of the only charge \k' concentrated at the point M', The proof of this lemma is immediate. CHAP. XII. - MURPHY'S PROBLEM. 267 Corollary. - If Vonjixe at the various points M', M", M'", . . . of the surface of the conductor Ci charges pi.', fx", ^" ^ . . . all positive, the sum of which is equal to Qi, these charges will induce on the conductor Co, put in communication with the ground, charges all negative, the sum of which will be equal to ( - Qo)? and we will have (1) Q2;=KQ" K is a positive constant, less than unity, and depends only on the mutual position of the two conductors Gj and Ga. It is first obvious from Lemmas ï and II that all charges induced on the conductor C2 are negative. The last part of our proposition therefore needs only a demonstration. A unit of positive electricity, in equilibrium with itself on the isolated conductor G2, brings it to the positive potential level A. Its potential function has, at the points M', M", M'", . . ., values A', A", A"^ . . . all positive, and lower than A. We have therefore A' Conductor Lo { . , ,/ I potential level ... H " . ,,, 1 charge O Point M' I . potential level ... B Point M" load O potential level ... B" This Gaussian identity will give us BQ, = BV-i- BV'-f- B'V'H-. ... All quantities in this equality are positive. Moreover ;jl' -h ;j." -I- [jl'" -4- . - 1 . The inequalities (2) give us Q2SKQ,. This is what we wanted to demonstrate. It would be shown in the same way that, if we fix at the various points of the surface of the conductor C2 charges all negative and whose sum is equal to ( - Q_-i), these charges will induce on the conductor Cj, put in communication with the ground, charges all positive whose sum will be equal to Q3, and Von will have (3) QsSK'Q^, K.' being a positive constant, less than V unit and depending on the mutual position of the two conductors C| and C:>. These propositions will serve to demonstrate the legitimacy of Murphy's method. They also provide us with information on the Gaugain coefficients (*) whose definition we will give. These coefficients are defined by means of three experiments. First experience. - The C| driver being maintained at the (' ) This theory is due to Mr. J. Moutier [.J. Moutier, Sur la condensation électrique {Bulletin de la Société Philomatique, 1878)]. CIIAP. XII. - THE PROBLEM OF MURPHV. 269 Potential level A, the C2 conductor is connected to the with the ground. The first takes a positive charge a, the second a negative charge - b. Let's put b - ma. The coefficient m depends only on the situation of the two conductors Cl and C2 and not on the potential level A. It is the first Gaugain coefficient. If the two conductors C|, C2 are outside each other, this coefficient is at most equal to K, and, consequently, less than unity. Second experiment. - The conductor Ci, maintained at the potential level A, is put in presence of the conductor C2, isolated and carrying a total charge equal to O. This one is put at the potential level B and the conductor Ci takes a total charge "q Gaugain calls the condensing force of the device the ratio "0 Third experiment. - We isolate the conductor C2, carrier of the load - b ei we put the conductor C| on the ground. It takes a positive charge"(. Let's put "1 = ni'b. The coefficient m' depends only on the situation of the two conductors C, and C2 and not on the load b. G isvXthe second Gaugain coefficient. If the two conductors C,, Co are outside each other, this coefficient is at most equal to K', and, consequently, less than unity. Gaugain found, through experience, that we had <4) a = "0 -+-"!- This result can be found by the theory. The Gaussian identity, applied to the first two experiments, gives ao A - a A - èB. This same identity, applied to the second experiment and to the 9-70 BOOK H. - DISTRIBUTION AND DIRICHLET PROBLEM. third, gives O = "lA - 6iB. These two relations give the equality found by Gaugain. As we obviously have a, = mm'a, the relation (4) gives 1 F - 1 relation that expresses the condensing force as a function of the two Gaugain coefficients. We will see later, in the study of the discharge of the capacitor, the importance of these two coefficients. § 3 - Murphy's method. Murphy's method has Lut to find the electrical distribution on two conductors C,, C2 ifig- 49)? one of which, d. is maintained at the potential level A, while the other, C2, is maintained at the potential level o, when one knows how to determine the distribution that given electric charges generate on each of these conductors, considered in isolation, and put in communication with the ground, or maintained at a given potential level. The conductor C, being at the potential level A and subtracted from the action of any external electric charge, is covered by an electric distribution that we can determine by hypothesis. Let Q be the total quantity of electricity it contains, a quantity which is positive if A is positive; let A| be the density of this distribution at the point M( of the surface Si of the conductor Ci; we know that this density A) is positive. Let us imagine this distribution yZ^ee on the conductor C); let us place in its presence the conductor C2, put in communication CHAP. X". - THE MURPHY PROBLEM. 27I with the ground. This conductor will be covered with an electric distribution that we know, by hypothesis, to determine; this distribution is monogenic and negative; ( - Q2) is its total mass; at a point Ma of the surface S2 of the conductor Co, it has a density (-A.) Let's fix this distribution on the C2 conductor and place it in the presence of the Ci conductor in communication with the ground. This one is covered with a distribution that we know how to determine by hypothesis. This distribution is monogenic and positive. Let Q3 be its total mass and A3 its density at point M, . Continuing in this way, we will determine, on the conductor C(, a series of monogenic and positive distributions having respectively for total mass Qi, Q3, Q", ..., and for density at point M, ^1, A3, As, On the Co conductor, we will determine a series of monogenic and negative distributions having respectively for total mass -Qî, -Qv, -Qs, ..., and for density at point M2 - A2, - A4, - Ag, The proposition stated by Murphy is as follows: If conductor C|, maintained at potential level A, is brought into contact with conductor C2 maintained at potential level o, the electrical density at point M, of conductor C) will have the value (5) 8, = A,-+- A3+ Aj-f-..., and the electrical density at the point Ma of the conductor C2 aut^a has the value (6) - §2 = - A2-Ai- Ae - .... This proposal, to which Murphy was obviously led 272 BOOK II. - DISTRIBUTION AND DIRICIILET PROBLEM. by the old theory of concealed electricity, is demonstrated by M. Cari Neumann (*) in the following way: Equalities (i) and (3) give Q2^KQ" Qv^KQ3, Qe^KQs, QaSK'Qa, Qs^K'Qi, , and, therefore, Qa^KK'Q,, QslKK'Qa, ..., The two series Qi + Q3-i-Q5-..., are therefore convergent. Let us now observe that we have Q,=:§Ai^S" Q3=§A3t/Si, Q5-§A,6?S,,, .. , Q^^^A^^S,, Qv-§A4^S,, Q6=gAcû?S2, and that the quantities A, , A3, A^, . . ., Ao, A,, Ae, . . . are all positive, and we will see immediately that the series (5) and (6) are convergent. These series thus determine a true electrical distribution on conductors C) and C2. H remains to prove that the potential function of this distribution is equal to A at any point of conductor C) and to o at any point of conductor C^. Let V" V3, \" ... the potential functions of the distributions A,, A,,, A" ..., and --V2, -V4, -Vfi,, ... the potential functions of the distributions - A2, -Ai, - Afi, (' ) Carl Neumann, Untersuchungen ùber das logarithmische und Newton'sche Potential, p. 3io. Leipzig, 1877. OHAP. XII. - THE MURPHY PROBLEM. 278 t is easy to see that the potential function of the distribution given by (5) and (6) will be given by the formula (7) "" --= V,- Vo-- V3_ V,-f- V,_ Vc^. . . . But, referring to the determination of the successive distributions, it is easy to see that we ?i, at any point of the conduc~ their C| , and, at any point of the Co, V, - V2=:0, V:,- Vi-:(), Vi- Vo=0, Formula (7) shows that we have, at any point of the conductor C|, V-= A and, at any point of the C2 conductor, r = o, which further legitimizes Murphy's method. § 4 - Combinatorial method of M. Cari Neumann. Murph's method solves, in a particular case, the general problem to which we have given the name Murphy's problem: knowing how to solve Dirichlet's problem for the space outside the conductor Ci and for V space outside the conductor Go, solve it for the space outside the set of conductors Ci and C2. Lipschitz (') and M. Cari Neumann (2) have shown how one can extend Murph's method}', in order to obtain the general solution of the previous problem. But, in spite of this generalization, Murphv's method leaves something to be desired in one respect: it has no analogue in the theory of harmonic functions of two variables (^) and, consequently, interrupts the parallelism between the theory of the (' ) Lipschitz, Crelle's Journal fiir reine und angevandte Mathematik, Bd LXI, p. 12. (' ) Carl Neumann, Untersuchungen iiber das logarithmische und Newton'sche Potential, p. 3i2. Leipzig, 1887. (' ) The corollary on which the justification of Murphy's procedure is based has no analogue in the p!an. D. - L 18 3.74 UVRE M. - DISTRIBUTION AND MANAGEMENT PROBLEM. logarithmic potential and the tliéoric of the Newtonian potential function. Mr. Cari Neumann (*) has re-established this parallelism by creating, to solve Murpliy's problem, a combinatorial method which also applies to harmonic functions of two and three variables. Let E be the space outside the two conductors C,, G2. H is to form a function V(M), hai^monic in the space E, and ])renant at the various points m, of the surface S| given values r(mi), and at the various points m-i of the surface S2 given values also (^(mo). The values ^^(m,) vary continuously on the surface S, and the values vi^m-^) vary continuously on the surface S^. Suppose we have determined a function W)(M), harmonic in E space, taking on the surface S| the given values ('(w,) and on the surface S2 the value o; then a second function Wo (M), harmonic in E space, taking on the surface S^ the given values (^(ma) and on the surface S( the value o. The function \Viai)^-\V2(M) will be harmonic in the whole space E, and will take the values given on the surfaces S|, So. The problem we seek to solve can therefore be reduced to two problems of the following type: Find a harmonic function in V space E, taking on the surface S, given values^ continuously variable, (^(/Wi), and on the surface S^ the value o. Let G be the largest absolute value of the quantities (^(/?î,). Let us form a first function, W,(M), harmonic in E space, taking on the surface S^ the value o and, on the surface Sj, the values, all positive, r(m))-|- G; then a second function, W2(M), harmonic in E space, taking on the surface S2 the value o and, on the surface S(, the constant and positive value G The function Wi(M) - WîCM) will obviously solve the previous problem. (' ) Carl Neumann, Untersuchungen, ..., p. 3i3. CHAP. XII. - THE PROBLEM OF MURPIIV. 7.yS We are thus brought back, in the final analysis, to the following problem: Find a function V(M), harmonic in the space E, taking at any point of the surface Sj the value o and, at the various points of the surface S, given positive values, <'(/??,), continuously variable. Let us form, which we know how to do by hypothesis, a function Vf (M), harmonic in the space (E + Co) and taking on the surface S| the given values p (/;?,). At the various points of the surface S2, this function takes certain values u\ . Let us then form, which we also know how to do by hypothesis, a function U2(M), harmonic in the space (EH-Ci) and taking on the surface So the values u]. At various points on the surface S, this function takes on certain values u'.^. Let us form a function U3 ( M ) , harmonic in the space (E -h G2) and taking on the surface Sj the values u'.^. At various points on the surface Sa, this function takes on certain values u"^. Continuing in this way, we will alternately form the functions contained in the two tables below: I. - Harmonic functions in space (£-+-€2). Values on Functions. S,. S^. U,(M) v(ini) u'\^ UsCM) u', u"^ U5(M) lù "5 II.- Harmonic functions in the space (E-f-Ci). Values on Functions. S^. S,. U2(M) u] u'.2 Ut(M) u":i U; Uc(M) u', '6 UyQ BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM. We will demonstrate that the equalized (8) V(M) = U,(M) - UîCM) + U3(M) - U^M) -^. . . defines a harmonic function in the space E, taking on the surface S, the values v{mt) and on the surface S2 the value o. According to the theorem of Mr. Vito Volterra, it is sufficient to show that, on the surfaces S^, So, this series converges uniformly to the values we have just indicated. Suppose that the function W(M), harmonic in the whole space (E + G2), takes the value i at any point of the surface S, ; it will take, at any point of the space (E + Go), values less than I . In particular, at any point of the surface So, we have the inequality WSK, R being a positive quantity certainly lower than the unit. Let us denote by G the largest of the positive values ç(m') given on the surface S). 11 It is easy to see that at any point of the surface So we will have u\ 1 KG. Let us consider the function GW(M) - Ui(IVl). It is harmonic in the whole space (E + Co); it is equal to o at infinity and cannot have any negative value on the surface Si; it cannot therefore take any negative value in the space (E -+- Co), and in particular on the surface So, from which the demonstration of the previous inequality follows immediately. We have therefore, on the surface S| , t^(m)Ç G, and, on the surface Sa, u\ ^KG. If we denote by K' a positive quantity, less than i and analogous to K, we can easily deduce from the previous equality that at any point of the surface S) we have u'^iKK'G. Continuing in this way, we will arrive at the results contained in the following tables: ICVP. XII. THE MURPHY PROBLEM. -^77 i" On' the surface Sj, takes the values positive function U,(M) v{m) UîCM) a'.^ U3(M) "3 = l<'., Uv(M) u U5(M) W'5 = î*4 U6(M) who have for liniile superior G KK'G KK'G K2K'2G K^K'îG K3K'3G 1° On the S^ area, takes who have The values for limit upper positive function Ui(M^ u\ KG UiCM) u".= u\ KG U3(M) ul K^K'G Uv(M) ul=ul K^K'G U5(M) ul ' K3K'2G U6(M) ul^u's K3K'2G Inspection of these two Tables immediately shows that the series which appears in the second member of equality (8) converges uniformly both on the surface S) and on the surface So; that it takes the value v(m) at any point m on the surface S, and the value o at any point on the surface So; this series therefore represents the harmonic function sought. This justifies the combinatorial method of Mr. Cari Neumann. It is hardly necessary to notice that this method lends itself indefinitely to extension. For example, let us consider three conductors C| , Co, G3. If we know how to solve the Dirichlet problem for the space outside each of them, we can, by a first application of Mr. Cari Neumann's method, solve it for the space outside the set of two conductors C|, C2; then, knowing how to solve it for this space on the one hand, and on the other hand for the space outside the conductor C3, we will solve it for the space outside the set of three conductors Ci, -278 BOOK I'. - DISTRIBUTION AND PROBLEM OF DIRICHLET. C2, C3, by a new application of the procedure of Mr. Cari Neuniann. We thus arrive at this beautiful theorem: Given n conductors outside each other in space, if Con knows how to determine the distribution that electricity takes on, on each of them alone, under the action of arbitrarily given fixed charges, we know how to determine the distribution of electricity on V all of these n conductors subjected to V action of arbitrarily given fixed charges. We know, for example, how to determine, in all possible cases, the distribution of electricity on a sphere subjected to a given influence. We will therefore be able to determine the distribution of electricity on two spheres outside of each other and mutually influencing each other. This beautiful problem was, in fact, solved directly by Poisson (' ), and has since occupied a great number of geometers, such as Plana, W. Thomson and G. Kirchhoff (-). (') Poisson, Second Mémoire sur la distribution de l'électricité à la sur/ace des corps conducteurs (read at the Académie des Sciences on September 6, i8i3). (") Foi> Mathieu, Théorie du potentiel. 2° Part, Applications, p. 65. Paris, BOOK m. EXPERIMENTAL STUDY OF ELECTRICAL DISTRIBUTION. CHAPTER ONE. THE BODY PROVES. § 1 - Theory of the body of test. After having seen how Poisson had, by means of questionable hypotheses which we will have to return to later, reduced the study of the distribution of electricity on conducting bodies to a problem of analysis, we have outlined the efforts by which geometers have attempted to solve this problem. We have now arrived at another part of our task; it is to take advantage of the analytical methods that we have studied, to deduce from them consequences that can be used for the experimental control of the theory of electrical distribution. The study of some of these consequences and their comparison with the facts will be the subject of the present book. We will begin by examining the method by which Coulomb experimentally studied the electrical distribution on a conducting body, the test body method. If a very small conducting body, carried by an insulating shellac needle, is electrified, the torsion balance in which this small body is taken as the fixed ball, while the moving ball is electrified by its contact with this small body, will easily allow to determine numbers proportional to the electric charges taken by this small body in various circumstances. 28o BOOK MI. - EXPERIMENTAL STUDY OF DISTRIBUTION. Therefore, it is easy to see that Ion will have a method to determine the distribution that affects the electricity on a conductor, if we admit the accuracy of the following proposition: A conductive body, of extremely small dimensions, being put in contact with another conductor of finite dimensions, takes an electric charge which is the product of two factors: i° the electric density which was at the point of the conductor which the test body touches before V approach of this body; i° a coefficient which depends on the shape of the test body and its situation with regard to the tangent plane to the conductor at its point of contact with the test body. This proposition was stated by Coulomb who sketched out a demonstration (' ). As we shall see, this demonstration can be given a more precise form, although some criticisms can still be made of its absolute rigor. This demohstration will be based on the following lemma ("): We consider two systems of homothetic conductors V one of V other, whose ratio of homothety is \. We suppose that the electric charges that these conductors carry in their homologous points are also in the ratio \. The potential function will have the same value at the homologous points of these two systems. If, instead of placing at the homologous points of the two systems charges which are between them in the ratio of homothety, we consider two systems containing only surface electricity, and if Von supposes that at the homologous points of these two systems the surface densities are the same, it is easy to see that homologous elements will carry charges which will be between them as the square of the ratio of homothety; at the homologous points of the two systems, the values of the potential function will be between them as the ratio of ho (' ) Coulomb, Sixth Memoir on Electricity. Continuation of the research on the distribution of the electric fluid between several conducting bodies; determination of the electric density in the different points of the surface of these bodies {Mémoire de l'Académie des Sciences for 1788, p. 676.) {') J. MouTiER, Cours de Physique, t. I, p. 4o6. ClIAP. I. - THE TEST MOVE. 281 moletia; the first order partial derivatives of the potential function will have the same values at the homologous points of both systems. This last consequence shows us that, if the electrical equilibrium is established on the first system, it is also established on the second. Gela posed, that is to say an electrified and isolated body G (/?^. 5o). Tl carries Fig. 5o. At each point, the surface density D is given. An isolated body E, in a neutral state, is brought close to it and touches it at the point P, occupying a given position with respect to the tangent plane TT'. The electricity is distributed in a certain way on the body E. Let A be its density at the point m of the surface of the body E. If we keep the body G invariant, but if, by homothety, we reduce the dimensions of the body E in a certain ratio "k, the center of homothety being in P, the density A tends to a certain limit d which we propose to determine. For this, let us take the state which corresponds to a certain value A of the homothety ratio. The body E has been transformed into a certain body e whose dimensions are those of the body E in the ratio À. Let be the value that the density A takes in this state. Let us form a system homothetic to this one, the homothety ratio being y The body e will become again the body E, but the conductor G will be replaced by a similar and larger conductor. If, on the latter, the density before the contact of the body E is the same as at the homologous point of the body G, we can easily see that after the contact the density will be the same at the homologous points of the bodies E and e. We can see that, instead of dealing with the original problem, we can deal with the following problem: without changing the 282 BOOK III. - lirCDE EXPERIMENTAL OF DISTUIBUTION. dimensions of the body E, we transform by homothety the body C, the center of homothety being in P and the ratio of homothety being a certain number [x which increases beyond any limit. It is assumed that the electric density before the contact of the body E is the same at two homologous points. We ask to what limit d the density A tends at a point of the body E after the contact. Now the limit state of the problem is as follows: An indefinite conducting mass is bounded by the unlimited plane TT'; before the contact of body E, the plane TT' is covered with an electric layer having at any point the density D(P) of the electricity which was at the point P of body C before the contact; body E is approached isolated and in the neutral state; what is, after the contact, the electric density d[m) at the point m of body E? We can admit that we have d{m) = Â-(m)D(P), k[ni) being a function of the position of the point /;?, a function whose form is fixed by the shape of the body E and by its position with respect to the plane TT'. From this expression of the electric charge in each point of our infinitely small test body, we see that the total charge Q taken by our test body will have the value Q=:KD(P), . K being a coefficient which depends only on the shape of the test body and its position with respect to the tangent plane at the affected point. This is the result we announced. Although, to determine the value of the coefficient K, one can arbitrarily dispose of the shape of the body C and the position of the point P on this body, the calculation of this coefficient can be carried out only for a small number of test bodies. Poisson ('), aj'ant solved the problem of electrical distribution on two spheres in contact, was able to determine the coefficient K for a test body formed by a small sphere of radius R. (' ) Poisson, Mémoire sur la distribution de l'Électricité à la surface des corps conducteurs {Savants étrangers, p. 62; 181 1). THE BODY OF KPRELVK. 283 It has Iroiivé K= -^4TrR2 4Tr3R5 Mr. Beltrami (' ) examined the case where the test body is formed by a hemisphere whose base BB' lies on the plane Fi". 5i. langent TT' {^fig- 5i). He found, by designating the hemisphere's rajon by R, K = 37iR2. Mr. G. Robin (-) treated a third case. The test body is the body of greatest attraction, whose polar equation is COS Ci I The constant a is the diameter; the body, which is of revolution around its diameter, presents at the pole an infinite flattening. We have then K = 27ra-. § 2 - Use of the test body. The calculation of the K coefficient, which is often very difficult, is fortunately unnecessary when one simply proposes to compare, at (' ) Beltrami, On the experimental determination of the electric density at the surface of conducting bodies (Il nuovo Cimento, 3" series, t. I, p. 2i5; 1877). (' ) G. Robin, Sur la distribution de l'électricité à la surface des conducteurs fermés et des conducteurs ouverts {Annales de l'École Normale supérieure, 2" série, t. III. Supplement, p. 9; 1886). ■28 1 LIVHE m. - KTIDE EXPERIMENTALALK OF THE DISTRIBUTION. by means of the same test body, the electric densities at the various points of the surface of the same conductor or of different conductors, without trying to know the total value of this density at each point. The preceding considerations are therefore sufficient to justify the use that Coulomb made of the test body in his research on electrical distribution (' ). This research has the particular interest of having provided the first experimental verifications of the theory, imagined by Poisson, of the distribution of electric fluid on conducting bodies. We have seen, for example (p. 191), that, according to this theory, at each point of an electrified ellipsoid, the electric density was inversely proportional to the distance from the center to the tangent plane at that point; this proposition is easy to submit to the control of experiment. Although Coulomb did not leave anything to this control, we know from the writings of Poisson (2) that he had tried it and found it satisfactory: "... This law, he says, was, in fact, verified by Coulomb on a wooden ellipsoid, covered with a metallic blade. This ellipsoid of revolution had been made on the lathe by President de Saron, an honorary member of our former Academy. " We have also seen (p. 209) that Poisson (^) had completely determined the electrical distribution on two conducting spheres, equal or unequal, placed in contact. On the other hand, Coulomb ('') made an experimental study of this distribution (' ) Coulomb, Cinquième Mémoire sur l'Électricité. On the manner in which the electric fluid is carried between two conducting bodies placed in contact and on the distribution of this fluid on the different parts of these bodies {Mémoires de l'Académie for 1787, p. 421)- - Sixth Memoir on Electricity. Continuation of the researches on the distribution of the electric fluid between several conducting bodies: determination of the electric density in the various points of the surface of these bodies {Mémoires de l'Académie for 1788, p. 617). (') Poisson, Mémoire sur l'attraction d'un ellipsoïde homogène, read at the Académie on October 7, 1833 (Mémoires de l'Académie, t. XIII, p. 5oi). (') Poisson, Mémoire sur la distribution de l'Électricité à la surface des corps conducteurs, read at the Académie on May 9 and August 3, t8i2 {Savants étrangers, p. 1811, p. I.) (' ) Coulomb, Cinquième Mémoire sur l'Électricité, ..., 1" Section, p. 437 CH.vp. I. - THE PROOF BODY. 285 whose results were compared by Poisson with the numbers deduced from his calculations. Coulomb put in contact two spheres whose radii were in a certain ratio b. After the contact, they were separated and moved away from each other. The electricity was distributed uniformly on each of them, taking on the smaller one a density [3 times greater than on the larger sphere. Here are the values of [S given by calculation and observation: Density ratio Electrical DifTerence Report of the shelves. on both spl icres following between the calculation and observation. the calculation. observation. b^\.... . p ^ r , I Go 1 ?-.,o8 - 0,07 b^l... . p^ i,3i68 ?. [,3o -*- 0,01 b=^l... .. 3-1,4443 P^.i,G5 - , l > "The first two differences fall within the limits of the errors of which observations of this kind are susceptible; the third, which is in the opposite direction to the other two, can still be attributed to these errors and proves nothing against the theory. However, it is good to observe that, in the case where Fun of the rays is the eighth of the other. Coulomb did not immediately determine the ratio according to which electricity is divided between the two spheres: to make the effect more sensitive, he made the large sphere touch the small one twenty-four times in succession; then he concluded from this complicated experiment the division of electricity in each contact. It is probably for this reason that the difference between calculation and observation is greater in the case of 6 = | than for other values of 6 (*). " Let's place two unequal spheres in contact. At the pole opposite the point of contact, on the "smaller sphere, will be the maximum electric density. Let y be the ratio of this density to the average density of electricity on the larger sphere when the smaller one has been moved away. The values of y observed by Coulomb and calculated by Poisson are contained in the following table: C) Poisson, loc. cit. p. 61. 286 BOOK m. - EXPERIMENTAL STUDY OF DISTRIBUTION. Difference y-values between the calculation Values - ^ and of b. calculation. experience. experience. b - \ ï = 1 ) 3'^^- ï = T , 27 -T- o , O4 h = \ Y = 1.834 Y =^ 1.55 -f-o,i5 h = \ Y = 2,477 Y = 2'35 -f- o,o"> Z> = I Y = 3.087 Y = 3.18 - o.o3 Two equal spheres being in contact, if we designate by h the electric density on one of them at 90° from the point of contact, the density will have for values at 180°, Co" and 3o°, a/<, a'/i, a"/i. A sphere S touches another one S' of double rajon. If we denote by ]i the value of the density on the sphere at 90° from the point of contact, the density on this same sphere at 180" and 60" from the point of contact will have the value [i/i, ^'A. At 90° from the point of contact, on the sphere S', this density will have the value ^' h. A sphere S touches another one S' of quadruple radius. If, on the sphere S, the electric density at the point of contact has a value h at 90° from the point of contact, on the same sphere, at 180" from the point of contact, it will have the value yA. Coulomb had determined by experiment the quantities a, a', a", [3, P', P", y. Poisson calculated them in his formulas. The following table summarizes the comparison between calculation and experiment: Ratio of electrical densities at different points of two spheres Differences which are affected between the calculation ^- - - ^ -^ - and calculation. observation. observation. a = 0.877 o' = 0.95 - 0.08 a' =1.342 a' =1.25 -+-0.07 a" =5.837 "" = 4.80 -h 1.06 1 P = 0.739 P = o.75 - 0.01 P'= 1.797 P'= 1.70 + 0.09 P"= 1.238 P"= 1.25 - 0.01 Y = 1.673 Y= 1:43 + 0.24 (:ini>. II. - OPEN CONDUCTORS. 287 CHAPTER I OPEN CONDUCTORS. § 1 - Electrical distribution on an open conductor subject to any influence. Coulomb (' ) was the first to state, as a consequence of experience, that electricity in equilibrium on a conducting body resides entirely on the surface of this body. As this proposition is one of the most immediate and general consequences of Poisson's theory, one would be in possession of an excellent verification of this theory if one could demonstrate that, in the equilibrium condition, there is no electricity inside the conducting substance. But it is easy to see that this truth cannot be demonstrated either by Coulomb's experiments or by any experiment. It is true that one can dig into a conductor cavities communicating with the exterior through small openings and prove that the walls of these cavities are not electrified; but the points inside the conductive mass itself are inaccessible and one cannot directly prove that they are in a neutral state. The experiments described in the Physics Treatises as suitable for verifying Coulomb's proposition are, in reality, experiments suitable for studying the distribution of electricity on open conductors^ that is, on conductors limited by two surfaces infinitely close to each other, bounded by the same contour that we call the edge of the conductor. Such is a spherical cap formed by a sheet of foil. It is therefore interesting to search in the study of the distribution (' ) Coulomb, Quatrième Mémoire sur l'électricité, in which two principal properties of electric fluid are demonstrated: the first, ...; the second, that in conducting bodies the fluid, once it has reached a stable state, is spread over the surface of the body and does not penetrate the interior (Memoirs of the Academy for 1786, p. 67). 288 BOOK III. - EXPERIMENTAL STUDY OF L/V DISTRIBUTION. electric on open conductors the explanation of the experiments in question; this explanation results from some beautiful theorems due to Mr. G. Robin (' ). Through the contour KK' {fig. 02), we pass an area limited to two sides S. At any point M of this area, we raise a normal on which we take, on both sides of the surface S, infinitely small lengths MM,, MMo. The points M,, Mo describe two surfaces S,, So, infinitely close to the surface S and passing through the contour RK'. J^the interval between the two surfaces S), So is filled by a conductive material. This conductor being charged with electricity and subjected to the action of given electric charges, the electricity distributes itself in such a way that its density is a, at the point M, and o-o at the point Mo. The problem of distribution on open conductors consists in determining towards which limits S,, So tend to o-,, 0-2, when the two surfaces S(, So tend towards the surface S. LetW(m)be the potential function at the point m of the surface S of all the acting charges. The potential function at the point /?i will be ^s, Ml m >^Sj M7 "i The surface S being entirely inside the conductor, this quantity must have the same value at any point m of the surface S. (' ) G. Robin, Sur la distribution de V électricité à la surface des conducteurs fermés et des conducteurs ouverts {Annales de l'École Normale supérieure, 2" série, t. III. Supplement, p. 9; 1886). CIIAP. II. - LliS CONDLCTELRS OPEN. "^89 The limit towards which it tends when the two surfaces S,, So tend to merge with the surface S will also have a value independent of the position of the point m on the surface S. This limit being obviously s \(/n) -+- \ =z=- we obtain the following proposition If, SU7' the surface S, one distributes an electric layer whose potential function, added to the potential function of the acting niasses, forms a sum which has the same value at any point of the surface S, and whose total mass is equal to the electric charge communicated to the open conductor, the density of this layer has the value (S, + So). This principle will be used to determine (S, + Ilo); this is how, for example, in Chapter VllI of the previous book we were able to determine this quantity for a spherical cap. (S, + S2) being assumed known, it remains to determine S, and S2. Let iN be the MM direction, and Ng the MMj direction. If an electric charge equal to the unit is placed at the point M,, it undergoes, from the layer of density o-, distributed on S,, an action whose component following N) is/, (M,); from the layer of density 0-2 distributed on So, an action whose component following N, is^2(M,); from the external charges, an action whose component following N, is 'i>(M,). We have, according to a known theorem, 27rscr.(M,)=/,(M.)-i-""'2(M,)-r- CllAI'. II. - THE OPEN CONDCOTEURS. 29I The second member of equality (i) has therefore the same limit as /i(Mi)+/2(M2)H-tî;(M2), and we have (2) 27:£[S,(M) - i:2(M)] = F,(M)-hF2(M) + ^f(M). Let q be one of the acting charges and Pie the point where it is concentrated. We will have S,(M'Uos(M 'M,N,) s rtm' S.2(M')cosrM 'M, Ni) 's W^\ 7cos(PM, N,) F.(M) = sC ^i(M-)cos^M'M,lN,)^^g^ '^s MM' F,(M) = .CME>^2liM2UV><,s, ^(M) = £"y ^*^ PM and the equality (li) will become [2.[x,(>i)_s,(M)]=y5L<^-^^i^^ I ^^ FM ^^ ' '^ [S, (M')-4-S2(M ')] cos(M'M,N,) I -S. s M'M We thus arrive at this theorem of Mr. G. Robin (' ): When Von has determined the sum (Si-f-Sj) of the electric densities at the corresponding points on the two sides of an open conductor, a quadrature suffices to determine separately the densities S| ei S2 at these two points. In the previous demonstration, we have confused, for brevity, the normals in M|, Mo to surfaces S|, So with (') If the surface S, is closed and the surface S^ is inside it, we have, at any point of the latter, a^= o; hence S^= o, and the equation (3) becomes ,... V 'ZCosCPM.N.) C\ S(M')cosfM'M.\ ) ,^ 2^-,(M)=: > ^ -2 - + \ =2 ^^'^- ^^ PM ^s MM This functional equation determines the electrical distribution on a closed confliicteur. Mr. G. Robin has made great use of this equation. He used it to solve the beautiful and difficult question of the electric distribution on a sjjheroid appreciably different from the sphere. Poisson had stopped at the case of the spheroid little different from the sphere (G. Robin, loc. cit.; see also, above, Book II, Chap. X). 292 BOOK III. - KTLDIi KVPÉRIMlîXT.VI.t: DE L.K UISTIUBI TIO.N. the normal in M to the surface S. This confusion is obviously made legitimate by the quasi-identity of direction of these normals, except at points very close to the edge RK' of the conductor. As, moreover, in the vicinity of this edge, the densities S,, ^o can be infinite, it is permissible to wonder whether the previous demonsti*ation is not thereby put in default. However, it is easy to see that the previous demonstration remains valid if we admit, as we shall, that at an infinitely small distance 8 from the edge KR' the densities 2l)i, S^, while being infinitely large, are infinitely small compared to < - In all known examples, they are of the oi'drc of - -- § 2 - Conductor o ivert removed from any influence. Let us suppose, in particular, that vm open and insulated conductor, to which a known charge of electricity has been communicated, is removed from any influence, and let us look for the electric distribution on a similar conductor. We know, first of all, that this distribution will be monogenous, and that the electric density will have in each point the sign of the total charge. Equation (3) will become (4) .,[i,(>l)-^.("i)] = Q f^.(^^^)H-I.(M^)^o.(M^^■.N,,^3^ ^s M'M Let us assume the surface S is everywhere convex in the direction of the normal N,. The second member of equality (4) will have a constant sign which will be that of [S, (M') + 112 (M')], i.e. the sign of the total charge distributed on the conductor. Hence this theorem: If an open conductor, whose surface is convex, is removed from any influence, the electric density has a greater absolute value at a point of the outer surface than at the corresponding point of the inner surface. If the open conductor carries a total load Q divided into a load Qj distributed on the face Sj and a load Qo distributed CIIVP. n. - THE OPEN CONDLCTEl'RS. 2Ç)3 on the Sa side, it is interesting to know in a simple way the limits between which the ratio ^- This result is obtained by the theorem that we will prove. Let us assume, for the sake of brevity, that the conductor opening is flat {/ig. 53), although the theories that will be expounded in Fie. 53. the study of curvilinear integrals (t. III, Book XIII) allow us to generalize the following proof. On the plane P of the opening, the edge. KK' cuts a bounded area A. Suppose that the surface S does not cut the plane P inside the area A. In the vicinity of the edge KK', the surface S is all on one side of the plane P; suppose that it is below this plane. The action exerted at the point M of the surface S by the electricity distributed on the surfaces S), Sj is equal to o; it is the same of the component following N) of this action and of the sum of these components for all the surface S; let us write the equality at o of this sum S.--.)[S cos(M',M, N,) ^S f/Si s Al", iVl J M'jM Let }JL be a point of area A and v the normal at this point to the upper face of plane P. The area A and the area S together form a closed surface to which the point M'j is external and the point M'^ -2'j4 LIVRK m. - EXPERIMENTAL STUDY OF DISTRIIÎLTION. integer. So, according to Gauss's lemmas, we have, n cos (M')M, Ni) ^^g _^ n cos(M'i;jL, y) ^^ ^ _ ^ -^s M', m' '^a M^' Scos(M',M,Ni) -^ n cos(M; [ji. V) c Ali' IVl ^-'i \l' . The previous equality then becomes 1 cos(M', u, v) fM l^S, ^à, L^A m;[jl' J Let us make the two surfaces Si and So tend towards the surface S; the two quantities ^A mY|I' ' ^A M^' tend towards the common limit ç\ ros(lM'[j., v) ^A, A M'fx' i.e. towards the angle to under which from the point M' we see the area A, this angle being counted positively or negatively according to whether the point M' is below or above the plane P. We thus have Let Q be the maximum and Q' the minimum of the values of w. The previous equality gives us JT.-iï ^ Qi ^ 4^-û ^^) - ?- >q;>-I> - These inequalities can still be written a' Os a (6) . >^< -. 4tc Q 4''^ CIIAP. II. - OPEN DRIVERS. 29" As an application of the previous inequalities, we propose to determine two upper and lower limits of the ratio-^ for a conductive cap resulting from the section of an ellipsoid of revolution elongated by the plane of a parallel {fig- 54) Let a be the angle of the base plane with the tangent plane all along the contour; 2 ^ the angle at the apex of the cone of revolution circumscribed by the contour from the pole of the cap. It is easy to see that we have ii = 2a, i2'=27r(i - cos^) and, therefore, cos3 Qî 2 ^ Q We see that when the dimensions of the opening are infinitely small of the first order, the total charge distributed inside the cap becomes infinitely small of the second order. A test body, put in contact with the interior walls of the cap, will not bring back any electricity. To the preceding theorems, M. G. Robin has added a large number of others; we will refer to his Memoir for the demonstration of the following ones, which we are only stating: i" A closed surface S {fg. 55) is formed by two caps C and C; if one knows how to determine, on the one hand, the influence exerted by any electric charges on the surface S; on the other hand, the influence exerted on the cap G by a charge placed in a 296 BOOK III. - KXPERIEN'TALE STUDY OF DISTRIBUTION. current point M' of the cap C, a triple quadrature allows to Fi g. 55. determine the influence exerted on the cap C by a load placed at any point in space. '>-" A closed surface S {^fig- 56) is formed by a zone Z and two caps C and C. Let us assume that we know: i" the Fig. 56. electrical distribution on the surface S subtracted from any influence; 2" the influence on the cap (Z + G) of a variable point M' of the cap C; 3° the influence on the cap (Z + C) of a variable point M of the cap C. From these data, we can conclude the distribution of electricity in equilibrium of itself on the zone Z. This beautiful theorem allowed M. Robin to deduce from the results obtained by Sir W. Thomson for the spherical callus the distribution of the electricity in equilibrium of itself on the spherical zone. CHAP. III. - LEVEL SURFACES AND ORTHOGONAL TRAJECTORIES. 297 CIIAPTER m. -VEAU SURFACES AND THEIR ORTHOGONAL TRAJECTORIES. § 1 - General theorems. The potential function of certain electric charges is, . a, which correspond to values of the potential function varying in arithmetic progression of infinitely small reason ( - a ) Let M be a point of the surface V; let ô be the normal distance of the l'-is;. 37. V+2aV+a> V V_a,V_ point M to the surface (V - "); let N be the direction of the normal to the point M to the first surface led to the second; this is the direction of the electric force at the point M, a force whose magnitude is F = - E £)N But we can, if we want, take c/N--o; then we have dS =^- - rt, and, [)ar consequently. The magnitude of the electrostatic force is, at each point, inversely proportional to the distance between the level surface that passes through this point and the infinitely close level surface. The surface of an electrified conductor is a level surface. Let us suppose that it corresponds to a certain value Vof the potential function. In the space surrounding this body, let us draw a level surface infinitely close to the surface of the body; 3oo Livniî m. - experimental study: of l.v distribution. it corresponds to a value (V - a) of the potential function, a being an infinitely small quantity, positive or negative. Let S be the normal distance from a point M of the conductor surface to this level surface. We will have at this point ÔN _ a But the surface density of electricity has the value at point M __ _ J_ dY_ 47t dNe' So we have a ~~47r8* The electric density is, in each point of an electrified conductor, inversely proportional to the normal distance between this point and the surface of level infinitely close to this conductor. Let's give, right now, an application of these general theorems. Let's consider an electrified ellipsoid removed from any influence. The potential function of this ellipsoid [Book II, Chapter VIÏ, equality (17)] depends only, in elliptical coordinates, on the parameter u. The level surfaces of this ellipsoid are therefore surfaces u = const. Hence these two proposals: The level surfaces of an electrified and uninfluenced ellipsoid are ellipsoids homofocal to it. The orthogonal trajectories are lines of intersection of the two families of homofocal hyperholoids at a given V ellipsoid. The general theorem proved last gives this proposition: At each point of an electrified and isolated ellipsoid, the electric density is inversely proportional to the distance between this ellipsoid and V infinitely close homofocal ellipsoid. CH.VP. ni. - DK LEVEL SURFACES AND ORTHOGONAL TRAJKCTORIES. 3oi Let's go back to the general properties of level surfaces. Consider an element on a level surface {fig. 58). Through all points on the contour of this element, let us conduct orthogonal paths. These lines generate a surface that bounds an infinitely unbound channel. Chasles, who pointed out the remarkable properties of such a channel, gave it the name of orthogonal channel; it is sometimes also called a force tube. Let's cut a similar channel by two normal sections c/o> and c/d)', sections which are obviously located on two level surfaces. The section thus formed is bounded by a closed surface to which we can apply Gauss' lemmas. If we suppose that the orthogonal channel has not met any electrified body between the two sections diù and d\ù we will have ^ F>" - dm -H V's diù' - G. From this equality it follows, in the first place, that F^ and FJ, are of opposite sign. If, therefore, the force enters the channel at one end, it leaves it at the other. The direction of the force in the channel therefore marks a constant direction of travel. 3uZ BOOK m. - EXPÉKIMENT.VLE STUDY OF L.\ DISTRIBLTIOX. Second, if we denote by F and F' the absolute values of the force at a point of the elements cIm and c/co', equality (i) can be written F doi - F' /- being the distance from point (^, J", z-) to pointM. Apply equality (i) to these two functions and we find S, ^s' Let us consider the third integral. Let dS be the angle under which, from the point M, we see the element û?S". We will have di." ^ R2 of, r \ [\ being the radius of the sphere. If this radius tends to o, we can easily see that the third integral tends to - 4'^V(M), and the previous equality becomes "' I ù\\... n i ,. "7, àV In the second integral, we have replaced the symbol N/ designating the normal to the surface S' towards the interior of the space considered by the symbol N^ designating, which amounts to the same thing, the normal towards the exterior of the surface S'. In the particular case where the surface S contains no acting charge, equality (2) becomes (3) '- OY ^^^'w=SA"^-^^/^^ CIl.VP. IV. - THE LEVEL COUCIIKS. 3o7 From equality (a), we deduce another analogous equality. Let us apply it, in fact, to the space between a closed surface S, outside of which certain acting charges may be found, and a sphere of very large radius. The term relating to the surface of this sphere will vanish when its radius increases beyond any limit, and we will have the equality, true for any point outside the surface S and the acting charges, '"^-<^')-SA^5^-^i;;''^-S. If all the acting charges are inside the surface S, this equality becomes simply (5) Let us examine the role of equation (3). When we know the existence, inside the space bounded by the surface i], of a harmonic function V, and when we know dY in addition the values of V and - - on the surface S, it allows to calculate the value of V at any point inside S. The functions of three real variables which verify the equation AV = o have properties which are very similar, in many cases, to those of functions of imaginary variables ('). The previous theorem is one of the main elements of these analogies. It is known that if a function y(5) of the imaginary variable z is finite, uniform and continuous within a certain area bounded by a closed contour 5 and if x denotes the affix of a point inside this contour, we have ^r.) ^l^fU-)r-..l f /-[^Idz, (' ) Among the important works for the study of these analogies, let us quote: !'. Appell, 5m/- the /onctions of three real variables satisfying the equation AF = o (Acta matliematica, t. IV, p. 3i3; i884). - P- Painlevé, 5m/- the singular lines of analytic junctions {Annales de la Faculté des Sciences de Toulouse, t. II. B.; i888). 3o8 BOOK III. - EXPERIMENTAL STUDY OF THE DISTRIBUTION. the contour s being traversed so as to leave the limited area on the left. The proof of this fundamental Cauchy theorem can be traced (') to the proof of equality (3) given by Green and reproduced above. This theorem shows that one can compute the values of the function f(z) at any point inside an area, if one knows the values of the function at the different points of the contour of this area. But there is an important difference between equalities (3) and (6). Equality (6) determines the function /(.r) inside the area when one knows only the values of this function at the various points of the contour. On the contrary, in order to calculate the value of V at a point in a certain space, equality (3) requires that we know in Ions the points of the surface that bounds this space, not not only the value of V, but also the value of its derivative -r^rr oN/ following the normal to the surface. Now, it is easy to see that this introduces superfluous elements (-) into the determination of V(M). As we have seen, the demonstrations given in Book 1, Chapter V, § 3, lead to the following result: For the harmonic function V to be determined unambiguously in all the space inside the surface S, it is sufficient that we know the values of V at all points of the surface S; or the values of V for certain regions of this surface and the values of ^^ for the other regions. If only the values of -rrr were known at any point on the surface S, the function V would be, inside this surface, determined to a constant. However, this apparent difference between the equalities (3) (;t (6) is transformed again into an analogy by a closer examination, because Cauchy's theorem requires, also, for the (') V. Hkrmitk, Cours d'Analyse de la Faculté des Sciences de Paris, written by H. Andoyer. {") G. KiRCHHOFF, Vorlesungen iiber matliematische physilc. Median il, , p. i85. ICVP. IV. - LAYERS OF xrvE.vi". 309 calculation of /(ar), superfluous elements. To determine /(x), it is not necessary to know the values of f{z) at all points of the contour, but only either the real or the imaginary part of f(z). In any case, the previous remark leads to this problem: Make disappear from the second member of the equalities (3) ,., . d\ . ,. and ( o) soil -t:-> or V. It is to make the second member of these eaa liles disappear that Green created the function we studied in Chapter VI of Book II, and, consequently, posed for the first time the problem which received the name of Lejeune-Dirichlet. F disappearing area V leaving only -r^ is the object of a similar problem, to which we shall return in Volume II of this work when we study the magnetic distribution, and which we shall call the Lejeune-Dirichlet derivative problem. Depending on whether we want to remove V from the second member of equation (3) or from the second member of equation (5), we find ourselves in the presence of the interior derivative problem, or the exterior derivative problem of Lejeune-Dirichlet. § 2 - Fundamental properties of the level layers. The exterior derivative problem of Dirichlet usually presents much greater difficulties than those presented by the Dirichlet problem. We shall see, however, that there is a special case where it can be solved immediately. Let's imagine that the surface S is an over/under level enclosing all the acting charges inside. Let A be the constant value of the potential function at its surface. Equality (5) will give, for any point M^, outside the surface S, But we know, by one of Gauss' lemmas, that the first of the two integrals in the second member has the value o. We have 3lO BOOK III. - EXPERIMENTAL STUDY OF DISTRIBUTION. so equality which solves immediately, in this case, the external derivative problem of Lejeune-Diriclilet. By which equality should we replace equality (7), when, instead of considering a point Me outside the surface S, we consider a point Mj inside the surface S, but outside the acting masses? We can always lead the surface S' in such a way that the point M,- is external to it; the equality (5) will then give Equality (2) will then become S, i4, [- \ )d.^ 0, or else, à cause of the cheerfulness to'r dz = 47:A, which respects Ite imm€ îdially d( he Gaussian lemmas, (8) Uv r dz^ .iTlA. Let us look for an interpretation, of equalities (7) and (8). We will name a level layer an electric layer distributed on the level 2 surface and having in each point a surface density determined by the formula I ^V _ _i_ ^ 4ti d\e "^411 àNe The potential function of this electricity will have as value at any point M in space the quantity U(M) = C ?<^X. CIUP. IV. - LEVEL LAYERS. 3ll Then, according to equality (8), at any point inside the surface S, (9) U = A, and at any point outside the surface S, according to equality (n), (10) U = V. It is concluded that a layer of level exerts, outside C the surface on which it is spread, the same action as the electric charges located inside V this surface and that it rC exerts no action in a point interior to this surface. Equality (9) shows that this layer, distributed on a conductor limited by the considered level surface, j would form an electric layer in equilibrium. Hence this new theorem: If Von knows how to find the level surfaces of a system of electrified masses, which contain these masses in their interior, we know how to find the distribution that would affect V electricity on a conductor limited by any of these surfaces. It is enough to give to the density in each point of this surface a value in inverse reason of the distance of this point to the surface of level infinitely close. This distribution admits the same external level surfaces as the charges originally considered. The total amount of electricity that forms the level layer has the value The Gaussian lemmas immediately lead to the following theorem: The total mass of a level layer has the same magnitude and sign as the masses whose action it can replace for the outer points. 3l2 BOOK III. ETCDE EXPERIMENTALE DE LA DISTHIBITION. These beautiful theorems were discovered by Grcen ('). Green's memoir was unknown when these propositions were found, almost simultaneously, by Cliasles (2), Ganss (-') and Sir W. Thomson (^). Complete study of an electrical influence case. The previous theorems lead us immediately to the solution of an interesting influence problem. ]^ bodies G, G',- G" are charged with electricity {fîg. 61); i] el ^' are two charge level surfaces distributed over G, G', G"; ^ envelops the electrified bodies and S' envelops '^. These two surfaces limit a conductive layer. N/, N^ are the dii'ections of ( ' ) G. Grken, An Essay on the application of matheniatical Analysis to the Tlieories of Electricity and Magnetisni. Art. l'i. Noltingham, 1828. - Grcen's Matheniatical Paper's, p. 63. (^) Chasles, Mémoire sur l'attraction d'une couche ellipsoïdale infiniment mince et les rapports qui ont lieu entre ces attractions el les lois de la chaleur dans un corps en équilibre de température {Journal de l' École Poly technique , t. XV, 25' Cahier, p. 3o4-3i6). - Énoncé de deux théorèmes généraux sur l'attraction des corps et la théorie de la chaleur {Comptes rendus des séances de l'Académie des Sciences, t. VIII, p. 209; iSSg). - Théorèmes généraux sur l'attraction des corps {Addition à la Connaissance des Temps for i845: published in 1842). (') Gauss, Alegemeine Lehrsàtze ûber die im verkehrten Verhàltnisse des Quadrats der Entfernung wirkenden Kràfte. Art 37 {Magnetische Verein; 1889. - Gauss Werke, Bd. V, p. 241). (*) W. Thomson, On the uniform motion of heat {Cambridge Matheniatical Journal; 1842. - Reprint of Papers on Elecirostatics and Magnetism, p. i el p. i3i). CIIAP. IV. - LEVEL LAYERS. 3l3 the normal to the surface -; N) , N^. are the directions of the normal to the surface S'. On the surface H', let us distribute a level layer; its density at each pointJ has the value _ j ^)r _ I d\' On the surface S, let us distribute a layer of level changed of sign; its density in each point has the value _ I 0\ _ I d\ Let be: U', U the potential functions of the two layers we have just defined; A, A' the values of V on the surfaces S, S'; P a point inside the surface i]; P' a point between the surfaces S, }i]';- P' a point inside the surface S'. According to the theorems of the previous paragraph, we will have : 1" At point P, (II) U(P) = - A. U'(P) = A': -^° Au |)oint P', {VI) U(P') = -- V(P'), U'(P')=:A'; 3" At point P", (iS) U(P')--- VC-P"), U'(P'') = V(P"). Let W be the total potential function, defined at each point by the equationC W = V -^' U + U'. i" According to the equalities (i 1), at point P, we have (14 ) W(P)-= V(P)-i-A'-A. 2** According to the equalities (12), at point P', we have (i5) W(P')-.A'. 3l4 LIVIIE m. - EXPERIMENTAL.VLK STUDY OF LV DISTRIBUTION. 3" According to the equalities (i3), at point P", we will have (i6) \V(P") = V(P"). Equality (i5) leads to the following theorem: i" The electrical equilibrium is established on the conductive layer limited by the surfaces S, S'. The properties of the level layers give iminediatomenl these propositions: 2" The layer o-' is equal in sign and in quantity to the charge OÏL spread on the conductors C, C, C"; the layer a is equal in size and of opposite sign to the charge SW. The equilibrium standard obtained on the conductor limited by these two surfaces is therefore the equilibrium state of this conductor supposed to be isolated and in the neutral state before having been influenced by the OÏL charges. 3° The layer i' would be in equilibrium of itself on the conductor which carries it. 4" The layer a- would be in equilibrium by itself on a solid conductor limited externally by the surface S. Equalities (i4) and (i6) still give this proposition: 5° The electrostatic action exerted either in a point of the enclosed space of the hollow conductor, or in a point of the space outside it, is reduced to the action of the inductive charges. § 4 - A particular class of capacitors. The properties of the level layers will also allow us to study in a complete way a particular problem of electrical condensation (' ). , If electricity were in equilibrium of itself on the conducting mass G {fig. 62), it would admit certain level surfaces; let S, S' be two of these level surfaces, the surface S' surrounding the surface S; between these two surfaces S, S', let us flow a conducting material C. We will have a system of two conductors (') J. MouTiKU, Cours de Physique, t. I; 1881. CHAI". IV. - THE COLCIES OF .MVKM:. 3i5 their of which one, C, bears the name ([''internal armature, the other, C, of^ external armature. The spherical Lejde bottle realizes a similar system; a cylindrical and very long Lejde bottle realizes it approximately. By placing one of the armatures in communication with a Fis. 62. If the source is at a constant potential level (') and the other armature is in communication with the ground, we will have a capacitor; this capacitor will take two different forms depending on whether the external or the internal armature is in communication with the source. i" The internal armature is in communication with the source at potential level A. The external armature is in communication with the ground. To determine the electrical distribution on the system, let us designate by V the potential level at which the conductor C would be carried by a charge equal to the unit in equilibrium of itself at its surface. The surfaces S, 2' would be level surfaces of this charge, and they would correspond to values u and u' of the potential function. Gela posed, let us distribute on the conductor C a charge a in C) The internal armature can only be connected to the source or to the ground if the external armature has a small hole. We will neglect the disturbing influence of this small hole. 3l6 BOOK III. - EXPERIMENTAL STUDY OF DISTRIIILTION. equilibrium of itself; on the surface S vme level curve changed sign of mass - 6 = - a; on the surface ^' let us place no electricity. Let Via be the potential function of the charge "5 let U be the potential function of the charge b. If we notice that the function V takes the value av at a point of the surface C and a value au at uu point of the surface S, we will easily see, from the properties of the level layers, that the total potential function has the value o at any point between the surfaces 2 and S', and a(v - u) at any point of the conductor C. If therefore we have been careful to determine the charge a by the equality , .% (17) a = > i^ - u we will have obtained the electrical distribution on the capacitor. The first coefficient of Gaugain [Liv. II, Ghap. XI] is defined by the equality ;;? := -- We have therefore, in the present case, (18) m - i. To obtain the coefficient m' , we must isolate the external armature with a load - b, equal here to - a, and put the internal armature in communication with the ground. This one will take a ciiarge "i and m' will be the ratio -j-' We can easily determine a,. On the surface of the body G, let us distribute a charge a, in equilibrium with itself; on the surface S, a layer of level changed of sign of total mass - a, ; on the surface S', a layer in equilibrium with itself, of total mass - 6 + "). It is easy to see that the equilibrium will be established on the system; that the conductor G' will carry a total charge - b\ that inside the conductor C, the total potential function will have a value "i c - ttiU - {b - ai)u . We will determine ", by equating this value to o. We will thus have (19) m - -y = ; j CH.VP. IV. - THE LEVEL LAYERS. 'ilj The condensing force of the device has the value I - mm or, according to equalities (18) and (19), (20) F = n ^- V - u ■.i° u external armature is in communication with the source at potential level =1. ; V internal armature is in communication with the ground. We will then have ( 17 OIS ) a =^ - , , II (18 /day) m = r > v -^ u - u (196/.V) m'.^i, {10 fjis) r - I H . V - u The condensing force of the device is the same in both cases. But the load taken by the collector is not the same. Let us apply the previous results to the case where the internal frame is a sphere of rajon R and the surfaces 2, S' are spheres, concentric to the first one, of radii p, p'. We have then The condensing force of the device has the value I I ÏÏ~p When we take the internal armature as a collector, it is charged with a quantity of electricity a = - I I ÏÏ-p I 1 u = -, u =: p P 3i8 LIVRB m. EXPERIMENTAL IDEA OF DISTRIBUTION. If, on the other hand, we take the external armature as a collector, it is charged with a quantity of electricity a - pM,. §5. Absolute electrometer by Sir "W. Thouison. The electrometer of Sir W. Thomson's electrometer may be compared to the capacitors we have just mentioned. However, we shall expose directly the theory of this device as the last application of the properties of level surfaces. Two conductors C, F are present (//^. 63). These two Fig. 63. conductors are partly located in a region of space bounded by a surface S, but they extend to distances from the surface S that are very large compared to the dimensions of this surface. Not only inside the surface S, but also at a great distance from this surface, the surface which limits the conductor C has the shape of two parallel planes P and P', the surface which limits the conductor F has the shape of two planes II, II', parallel to these two. Conductor F is connected to the ground and conductor C to a source at potential level a. It is easy to find the electrical distribution on these two conductors - we are only interested in the distribution on conductor F. Within the surface S, the level surfaces can obviously differ only slightly from parallel planes. Let x be the CUAP. IV. - THE LAYERS OF MVE.VU. 3 KJ dislance of a point from the plane P. The Laplace equation, which the potential function must verify outside of the acting masses, is then reduced, for those points which are inside the surface S, to the form dyy _ If one designates by D the distance OA of the two planes P, II; by E the thickness 1111' of the plate F, one sees without difficulty that, in the space included between the planes P and D, one has (21) \- -x^o. In the space beyond plane II', it is easy to see that (29.) V = 0. Let N be the exterior normal to the conductor F at a point on the plane n. Let N' be the exterior normal to the same conductor at a point of the plane n'. Equality (21) gives 5N ~ 1)' and equality (22), dy_ _ dN' ~ ^' The plate F does not carry electricity on its face II'. On its face n, it carries an electric layer whose density at each point has the value (7 Let's suppose that a cylinder MM'NN', normal to planes II, II', divides the conductor F in such a way as to isolate a moving disk. In order that this disk and the remaining part of the conductor F (guard ring) continue to form a single conductor, electrified as we have just indicated, a wire F joins the disk to the guard ring. Let 2 be the base surface of the mobile disk. This disk, carrying electricity only on its lower base, is subjected to a force, directed along N, and having for magnitude 320 BOOK in. - EXPERIMENTAL STUDY OF DISTRIBUTION. In other words, the moving disk is attracted by the fixed disk, and the attraction has the value ^^8^ï^"' It will suffice to balance this attraction by a known force, to obtain the determination of the absolute value of the potential level. CHAP. V. - THE GREEN PROBLEM AND FARADAY'S THEOREMS. 32t CHAPTER V. THE GREEN PROBLEM AND FARADAY THEOREMS. § 1 . - Green's inner problem. - Green's solution. The properties of level layers give the immediate solution, in an extended case, of the following problem: Given electrified masses and a surface surrounding these masses, distribute on this surface an electric layer exerting, at any point outside the surface, the same action as the given masses. This problem was first tackled by Green ('); Green showed that this problem could be solved when one knew how to find, for the space inside the given surface, the function to which Riemann gave the name to Green's function. Let's prove, first of all, that the problem we have just stated cannot admit several solutions. Let us suppose, in fact, that on the same surface, two distinct layers have been distributed, having for density at point M one a-, the other t'. Let V be the potential function of the first distribution and V the potential function of the second. The two distributions exert the same action at a point outside the surface; therefore, outside the surface, the two functions V and V admit the same partial derivatives. If we add that they are both equal to o at infinity, we see that at any point outside the surface, we have V-V'=o. (' ) G. Green , An Essay on the application of mathematical Analysis to the theories of Electricity and Magnetism. Arl. 5. Nottingham; 1828 {Green's Mathematical Papers, p. 3i). D. - I. 21 322 BOOK III. - EXPERIMENTAL STUDY OF DISTRIBUTION. The two functions V and V having the same value on the surface in question and being both harmonic inside this surface are identical between them inside this surface. We have therefore, in all space, the identity V - V = o. Let N/, Ne be the interior and exterior directions of the normal in M to the given surface. We will have OR, by virtue of the previous equality, This is what we announced. Assured thus of the equivalence of all the solutions of Green's problem that can be obtained by various procedures, let us first expose Green's solution. First, we propose to solve Green's problem for the case where the surface S contains only one electrified point P {/ig'- 64 ) carrying a charge /?z. Fig. 64. Let G(M/) be the value at the point M/, inside the surface S, of the Green's function of pole P. We know that, if /* designates the distance M/P, we can write (Liv. II, Chap. VI, §2) G(M,-) = r(M,) + ;J:, r(M/) being a harmonic function inside the surface S and equal to ( ) on the surface S. CHAP. V. - THE GREEN PROBLEM AND FARADAY'S THEOREMS. 3^3 This being so, let us consider a function V(M) thus defined: 1° At any point M/, inside the surface S, we have V(M,)=-/nr(M,); 2° at any point M^, inside the surface S, we have The function V(M) is continuous in all space, equal to o at infinity, harmonic outside the surface S, harmonic inside the surface S; it is thus the potential function of an electric layer distributed on the surface S with density This layer having, at the points outside the surface S, the same potential function as the mass m, solves Green's problem for the case where the surface S contains a single acting mass. If the surface S contains several acting masses, we will solve Green's problem by superimposing the distributions which would solve this problem for each of the masses taken in particular. We can see that we know how to solve Green's problem if we know how to find Green's function for the space inside the surface S. It is easy to recognize the exactness of the reciprocal. § 2 - Gaussian solution. Without knowing Green's Memoir, Gauss was led, for his part, to ask himself the same problem (' ); he was the first to propose a general demonstration of the existence of one and only one solution to this problem; according to what we have just said, it is there, by (') Gauss, Allgemeine Lehrsâtze ùber die im verkehrten Verhàltnisse des Quadrats der Entfernung wirkenden Kràfte, Art. 29-34 [Magnetische Verein, 1889 {Gauss Werke, Bd. V, p. 281)]. 324 BOOK III. - EXPERIMENTAL STUDY OF DISTRIBUTION. as a counterpart, a demonstration of Texistence of Green's function and of the correctness of the so-called Lejeune-Dirichlet principle. This demonstration by Gauss is not free of criticism; nevertheless, it presents fewer contentious points (' ) than the demonstration of Lejeune-Dirichlet, reported in Book II, Chap. V, § 2. We shall therefore explain this demonstration here. Theorem I. - A surface S {fi g- 65) contains in its interior electric charges whose potential function is U. One can always, on the surface S, distribute a mo nogen of given total mass 011, so that the difference between the potential function V of this layer and the function U has the same value at all points of the surface S. Let us consider, in fact, a monogenous distribution, of given mass 31i, on the surface S; let us suppose, to fix the ideas, that this distribution is positive. Let u be the density of this distribution at the point M of the surface S. The quantity V will certainly be positive at any point of the surface S. The same will be true for the quantity This quantity is therefore limited to a smaller amount. The same applies to the quantity - CaUcrcfô, (' ) See, on this subject, Carl Neumann, Untersuchungen iiber das logarithmische und Newton'sclie Potential, p. XI and p. 107; Leipzig, 1877. CHAP. V. - THE GREEN PROBLEM AND FARADAY'S THEOREMS. 325 because the latter can never become less than X being the largest of the values of U on the surface S. The quantity Q= C(V - 2U)c7o. If the external forces which solicit the system admit a potential, this proposition can be transformed into the following one: For V equilibrium of the system, it is necessary and sufficient that the value of the total thermodynamic potential is a minimum among all the values that this quantity can take at the same temperature. The internal thermodynamic potential has a large number of properties, some of which will be useful to us later. i" Assume the state of the system defined by the temperature T and by a number of independent parameters a, , ao, . . . , a^^. CIIAP. r. - NOTIONS OF THERMODYNAMICS. 34 1 Let's put To maintain the system in equilibrium in the state defined by the value system ai, (Xo, ..., %n, T, it is, by hypothesis, necessary and sufficient to place it in an enclosure at the same temperature as it and to apply certain external forces to it. These forces must be such that the work done by them in any isothermal modification of the system has the value Pi 5ai -f- P2 072 -H . . -^- P" 02t" In particular, to maintain in equilibrium a homogeneous body whose specific volume is r, it is necessary to apply a normal and uniform pressure whose value is 2° Let us suppose that the parameters a,, ao, ..., a^ are chosen in such a way that, if T varies, ai, ao, ..., a" remaining constant, no external work is done, and no live force taken by the system. This is what will happen if all the parameters on which the shape of the system depends are among ai, ao, ..., a". We will then have (9) ^=-^^' (10) ,:?-T^ = EU. 3° In these conditions, let g?Q be the quantity of heat released by the system during any isothermal modification. We will have (") ErfQ = -3(.f-T^f)+^5.-S2=^ In the particular case where the modification is reversible, we have 8> = o, oj + c?fce=o. 34-;i BOOK IV. - THE THERMODYNAMIC POTENTIAL. and the previous equality becomes simply (,.) E "rQ = a(T^). 4*^ We will often give the name of compensated work done in a modification to the quantity (i3) t^& = - ET8S. If we put (i4) H=ETS, in any isothermal modification, we will have (i5) 3" Of the second group of added forces; in the considered modification, these forces perform, by hypothesis, the work of^]. CIIAP. I. - CONCEPTS OF THERMODYN.VMIQUK. 345 So we have f/8 = d<£e - divg -+- dxB'i = d(E'i, and equality (19) becomes dz = - E 0(1] - TS) -h dS'i. According to the inequality (18), this quantity is zero or negative; therefore, according to the propositions of Thermodynamics that we recalled at the beginning of this paragraph, the corresponding modification, leading to zero or negative uncompensated work, cannot occur; the system, subjected to the mechanical actions that really act on it and to those that we have added, cannot undergo any displacement that does not alter the state of its various parts; the connections that forbid any modification other than displacements of this kind ensure the equilibrium of the system. But, thanks to these connections by which each of the bodies that constitute the system is supposed to be maintained in an invariable state, the propositions of Rational Mechanics relating to systems formed of solid bodies are applicable to the preceding system. In particular, the principle of virtual velocities can be applied to it. From the above, the system is in equilibrium under the action of four given force systems: 1° The external mechanical forces, which acted primitively on him; 2° The internal mechanical actions, that the various bodies which constitute it exert on each other; 3" Equal and directly opposite forces; 4° Forces performing, in any virtual displacement, a work cKb'^ equal or lower than Eô(U - TS). The first and third group of forces destroy each other. We can therefore say that the system is in equilibrium under the action of the second and fourth group of forces. But, in any modification of the system where its various parts change position without changing state, the forces of the second group perform a virtual work d(Ei; the forces of the fourth group eff'ect a virtual work d(B'^; by virtue of the principle of 3^6 BOOK IV. - THE THERMODYNAMIC POTENTIAL, virtual velocities, the system cannot be in equilibrium under the action of these two groups of forces if we do not have d(s>i -+- d^'i le. This must take place whenever d^\ verifies the inequality (i8). We must therefore have 6/S,-^- E8(U - TS). If we now suppose that to any virtual displacement of the various parts of the system we can make the inverse displacement correspond, we will easily see that the preceding inequality reduces to the equality (17), i - say, which can be stated as follows: When a system undergoes a displacement without change of state of its di^'crses parts, the uncompensated work produced in the system is equal to the work produced by the given mechanical actions, both external and internal, acting on the various parts of the system. CorollaryW. - We have seen that the entropy of the system was related to its internal tliermodynamic potential by the equality (9), ES - - . If therefore d^ designates the compensated work that accompanies a CHAP. I.- NOTIONS OF THERMODYNAMICS. 847 modification without change of state, we have But ( - ùj) can, in this case, be replaced by the variation that the potential of the internal mechanical actions of the system undergoes. If these mechanical actions are independent of the temperature, it will be the same of their potential and we will have, consequently, /y, assume that this charge is uniformly distributed in volume vi. Gela posed, it is easy to demonstrate that the quantity /y depends neither on the shape of the surface which bounds the volume Vi, nor on the orientation of this surface with respect to that which bounds the volume Vj. Whatever the shape of the surface which limits the volume p/, q,uatever the orientation of this surface with respect to that which limits the volume Vj^ we can always subdivide the volume vi into an unlimited number k of infinitely small cubes ", b, . . . , l, all equal to each other and to an infinitely small cube taken as type, having all, with respect to the surface which limits Vj, the same orientation independent of the orientation of the surface which limits Vj. l, all equal to each other and to an infinitely small cube taken as a type, all having, with respect to the surface which limits Vj, the same orientation independent of the orientation of the surface which limits vj. Let (^aj be the quantity analogous to ^ij relative to the cube a and the volume çj. It is obvious that we must have */y = ^aj H- Ç/v + -..-+- O/j. The amount of electricity contained in each of the small cubes a, b, ... ^ /has the value ^- It is the same for all. The parameters which define respectively the systems a, /; b, j; ...; l, / have therefore the same value or approximately the same value for each of them; therefore, we have approximately and 11 It is easy to see that the second member does not depend on the CHAP. II. - THERMODYNAMIC POTENTIAL OF AN ELECTRIC SYSTEM. 35 1 the shape of the surface that limits the volume P/, nor the orientation of this surface with respect to the surface that limits çj; the stated proposition is therefore demonstrated. We would also see that $/y does not depend on the shape of the surface that bounds t'y, nor on the orientation of this surface with respect to the surface that bounds c/. Let /;//, nij be the masses of the two parcels ç^/, vj; let T be their temperature; let a, [3, ... be the parameters, such as the densities, which fix their physical and chemical state. From the above, we will have "ï>/y= *(m,, /n;, Ci, qj, nj, T, a, p, . . .). We will show that the quantity ^ij is a linear and homogeneous function of the niasse nii of the parcel vi and the electric charge qi it carries. Let us divide the volume Vi in any way into a number of equal or unequal volumes ",6, . . . , /. Let m^, m^, . . . , m/ the masses of these elements; let qa-, qt-, - - - , <7^ be the electric charges they carry. For all these elements, the parameters ï, a, p, ... are approximately the same; we have also approximately raj = r/,j - ...= nj = rtj. We must therefore have *(m/, qi) =i^(/na, qa)-i-^{mb, q/,) H-... -H ^{ni/, q/), equality in which we have highlighted only the parameters that do not have the same value for all elements. But : 1° The masses ma-, mi,, ..., mi are subject to this single provided that nia -{- ntb-r. . .-+■ mi= mi; 2° Since the distribution of the charge qi on the element Vi could be chosen arbitrarily, we see that /y is a linear function and homo 35jt BOOK IV. - THE THERMODYNAMIC POTENTIAL. £>ene of the mass mj of the element py, and of the electric charge qj that it carries. The results we have just obtained are expressed by the following formula (3) *,y= mimjOijirij) -\- mjgi'i^ij{rij) -+- miqj^ji{rij) + q/gjyjjirij), in which it should be noted: i" That the functions cp, y, "]; can change their form when the parameters T, a, ^, - . . change their value ; 2" That one has well, by reason of symmetry, but not It is easy to see that, to form the quantity ^^', given by equality (2), we can, by virtue of equality (3), operate as follows. First, for the volume Cj, we will form the quantity [§\= - /ni["ï2 9i2(''i2)-i- "Î3 ?i3(^i3) + - .-4- /n "çi"(/-,")] (4) I + qiV'n^'^iiirii)-^ nii<]^],z{rii)-^. ..-^m,i<]/ui{rin)] f ■-+- - 9'i[5'2Xi2(''i2) + S^s/isC^'ia) -H...+ ^"Xi/i(ri")]. Then we will form the same for the volumes c'2, <';;, ..., v,, the quantities §'^, ^'^, . . . ., éF)^. We will then have (5) .r= #1 + J'2 + ---+^/r Suppose that the volume element Vi is part of a body of which A' is the density at the point of coordinates {x, z'). Suppose further that the electricity is spread inside the body and that p' is its solid density at the point {x, z'). We will have, by virtue of equality (4), ♦ ,fi= Wi / / / àJ oi^r) dx' dy dz' -^ Çi f I f ^' '^ {r) dx' dy' dz' \^^JJj ?' X{'') dx' dy dz' . CHAP. H. - THERMODYNAMIC POTENTIAL OF AN ELECTRIC SYSTEM. 353 Each of the three integrals \] ^ Ç f Ç \'o{r)dx' dy'dz', (6) i \ = f Ç f i^' <^ {r) dx' dy' dz\ f W = f fp'y.ir) dx' dy' dz\ in which /- denotes the distance from the point [x',y',z') to a point of the element <,, must exist-, which requires that each of these integrals, extended to any uniquely small volume including a point M in its interior tends to o when this volume somehow vanishes at point M. Once this has been done, let us come to the determination of the quantities y*),y2, ...,y", which appear in equality (i). The quantity fx does not depend on the Jorme of the surface that limits the volume t^j. To prove this first proposition, we will rely on a lemma that follows immediately from the proposition we have just established. Let us imagine that we divide the element t'i in any way into an infinity of other elements ", b, . . . , /, of volumes "a, Ubi - - - 1 ui. Let/rt be what becomes the quantity y, for the element a. Let Ua, \ai Wa be the values, at a point of the element a, of the integrals given by the equalities (6), these integrals extending to the volume V). It is easy to see that we will have -^ qaya-\- qb^ b-'r ... .-^ qi\ t-^ 7 (<7a\Va + C'A Wô-t- ... -+- 5-/ W/) ; but Ua, U,, ... U/, V", V,, ... V/, W", W/" ... W/ are, according to what has just been demonstrated, infinite quantitiesD. - I. "3 354 BOOK IV. - THE THERMODYNAMIC POTENTIAL. small as <). We can therefore write (7) /i = /a + /^. + ... + //. With this lemma, let us prove the stated proposition. The distribution of the charge q^ on the volume V is indifferent, so let's suppose it is uniform and let's divide the volume V into A" small cubes ", ^, . . . , / equal to each other and to a cube taken in advance as a type. These cubes will be identical. We will have then and, by virtue of equality (7), The number k depends on the volume of the element (^,, but not on its shape; the same is true of /"; therefore, the same is true ofy*,, which had to be demonstrated. The quantity f^ is a linear and homogeneous function of the mass mt and the load q^ of the clothing p,. To prove this proposition, we divide in some way the element C) into k other elements a, ^, ...,/, having masses m^, m^, . . . , m/ and carrying charges ^a, qi,^ . . . , qi. Equality (7) will give us, by only highlighting, among the parameters on which y, depends, those which vary from one element to another, /i(m,, qx)^fx{ma, qa)^fï{r)ib, Qh) ^ - . .-^ fi{>n,, g/). But we'll note: i" That the masses m", mi, . . . are subject to the only condition that we have 0.° That the distribution of the charge ^, on the element ç, being arbitrary, we can always dispose of the charges qa-, qb, - - - -, qi-, subjecting them to this single condition <7a-+- ^6-*-----H qi- q\ Hence, the resulting equality demonstrates the previously stated proposition. CHAP. II. - THERMODYNAMIC POTENTIAL OF AN ELECTRIC SYSTEM. 355 In summary, we have (8) /i= niiFi-hqiGu F,, G) depending on the temperature of the element c, its density and the parameters that determine its physical and chemical state. The set of equalities (i), (4), (s), (8) determines the form of the thermodynamic potential of an electrified system. § 2 - Introduction of the fundamental hypothesis of compressibility. We have seen that, if we move the various parts that make up a system relative to each other without altering the state of each of these parts, the forces given inside the system perform a work d^i related to the variation of the internal thermodynamic potential by the equality (9) d(Bi^ - Brf. Let us apply this theorem here. Let us displace with respect to each other the volume elements Pj, (^25 - ■ --, ^n into which we have decomposed an electrified system, without changing the state of these various elements. We will have, from the equalities (i), (4), (5), 8^ = 8#'i -t- S^'j + . . . -f- 8 j; and 2 L '''■lî ^'13 ^''in J Equality (9) then leads to the following consequence: The given internal forces acting in a system whose various parts are electrified fall into three classes: \Two material masses /??,, m^, electrified or not, exert an attraction on each other /"i nii ' . orn 356 BOOK IV. - THE THERMODYNAMIC POTENTIAL. proportional to the product of the two masses, depending on their distance and their physical and chemical state; this force is subject to the rule of Inequality of action and reaction. 1° Two electrified masses, carrying charges ^,, ^o, exert V one on Vautre an attraction <(Ji - -, proportional to the product of the two charges, depending on their distance and their physical and chemical state; this force is subject to the rule of equality between action and reaction. 3° A material mass m^ exerts on an element of volume carrying an electric charge ,. Now this pressure is given by the equality nii .P.,..),, nii {-'t- 771 1 i^'t--' oAi SA] = G. For it to have the same value as in the previous case, it is necessary and sufficient that we have '"^Ir-^^^/AT^"' whatever the quantities m^ and ^2" Lies two functions 'fi^, (^7-12 the function cp,2 being able to change form with the physical and chemical nature and with the temperature of the two molecules m, and mi^ but not being able to depend on their density. 1° A material mass nit and an element carrying an electric charge q^ exert a mutual attraction the i2i carrying electric charges Çi, q-y, are located at a distance /■^2 greater than a certain very small length \ which we will name the radius of molecular activity, their mutual attraction reduces to As a result, when /"ja surpasses )>, the quantity ^(2 differs only from by a term independent of /(o, and this term is zero, since d/,2 must cancel when /(o becomes infinite. We have thus Xl2(''l2)= ;r-®i2(/-12), '12 the quantities Si2i ^a, '^{2 depending on the physical and chemical state, on the temperature of the elements Ci, Co, but not on their density in the case where we admit the hypothesis indicated in the preceding paragraph. These quantities become equal to o when /'la exceeds 1. Finally, let us add this hypothesis: The sum ©12(/'l2)'/2-i- ''12 ''13 f'in the sums being extended to all the elements of the system other than the element Vt. Let us then pose (i2) Pi= 2Fi + £i(rn)m2-^£n{ri3)ni3-i-...-h£in{rin)>nn, (i3) 01= Gi-+- t{^i(ri2)m2-4-i3(/'i3)/"3-t-----i-4'i"(''i")"^"" the sums extending to all elements located in a sphere of radius A having its center in a point of the element (^|). We will have, by virtue of equalities (i), (4), (5), (8), £ = - - ( mi iii -f-"12 i22 + - - -+" "^n^n ) , , , + -(g-i Vi-(-72 V2-+-...+ ^,i V") (>4) i -^ + - ( mi Pi H- /?l2 P2 + . . . -r- nin P/i ) Each of the four summations in the second member extends to the entire system. Each of the four terms in the second member has a remarkable meaning. The internal actions that are exerted in an electrified system are, under the assumptions made, of four kinds: 1° Two material masses exert on each other an attraction given by the law of universal gravitation; the potential of these actions is the term (niiQi -\- m^Qi-h . . . + mn^n) The study of this term constitutes Celestial Mechanics. 362 BOOK IV. - THE THERMODYNAMIC POTENTIAL. 2° Two material masses, located at a distance less than the radius of molecular activity, exert on each other an attraction that depends on their physical and chemical state and their temperature. These forces, called capillary forces, have for potential the term - (miPi -\- m2P2 + . .-I- /n "P"). 2 The study of this term constitutes the study of capillarity, in the general sense that we can give to this word today. We have sketched elsewhere (^) how Thermodynamics could bring out from this starting point the whole study of capillarity. 3" Two electrified masses exert on each other a repulsion given by Coulomb's law. These forces have for potential the term -(g-iVi-^- <72V2-t-...H-gr„V„). 4° Two masses "^^, m^, carrying electric charges ^,, ^2? and located at a distance smaller than the radius of molecular activity, exert on each other an attraction ()4^i2(ri2) to<]t.2i{ri2 + ...-+-/n "a") -I- - (m, Pi -i- m,P,+...H-m "P"). On the other hand, let W be the electrostatic potential of the system; we have With these notations, equality (i4) becomes (i5) i = E(r - TS)H-\v+e,^,+ e2^2 + -.- + Q "7". ■ This is the form of the internal thermodynamic potential of an electrified system which we had already arrived at in 1886 (^). The formulas (9) and (10) of the previous chapter immediately give us the internal energy and the entropy of an electrified system. If we put AT (-7) K^e-T^, (' ) II. Hklmholtz, Ueber die Erhaltung der Kraft, p. 4? (Berlin, 1847Helmholtz, Wissenschaftliche Abhandlutïgen, t. I, p. 48). (' ) P. DuHEM, Le potentiel thermodynamique et ses applications, p. 209. 364 BOOK IV. - I,K THERMODYNAMIC POTENTIAL. we will have (i8) ES =ES-i-IIi^,-f-Il2 5'2H--..+ H"^", (19) EU= Er + W-4- K,^i-i-K2^2 + ...-(- K"^". In what follows, we will often have to use these formulas (i5), (18), (19). § 4 - On the continuity of the quantity e. By focusing on the principles we have just explained, we will arrive, in the next chapter, at this consequence: The quantity (i3) e, = Gi+ 4/12(^12) m2-+-4'i3(n3) "î3-)-. ■. + ^\n{rxn)inn varies continuously with the coordinates x^^ j^,, z^ of a point of Vêlement v^^ even when this point crosses the surface of discontinuity which separates two bodies of different nature. Let us see, right now, what the correctness of a similar ])roposition implies about the quantities G and <^^^. We have seen (§ 1) that, if A' is the density of the element dx'dy'dz'^ the integral /// A' (^ ( /') dx' dy dz' , where /- denotes the distance from a point of the element dx' dy' dz' k to a point of the element (',,, and which is extended to an infinitesimally small volume comprising the element Vi, must be infinitesimally small regardless of the shape of that volume. Based on this lemma, we can easily show that the previous proposition is indeed true if we admit the two following hypotheses: 1** The quantity G, is an absolute constant which does not depend on any particular property of the element Vi. 1" The quantity (|>,2, while depending on the physical and chemical state of the temperature of the mass m2, does not depend on any of the properties of Vt which carries the mass q^ . CHAP. II. - THERMODYNAMIC POTENTIAL OF AN ELECTRIC SYSTEM. 365 We will admit these hypotheses, the second of which is in accordance with Hehnholtz's way of seeing as explained at the end of the previous paragraph. We will also admit, in what follows : i" That the first order partial derivatives of quantity 6 vary continuously in all space; 2° That the second order derivatives of the quantity 6 only experience discontinuity at the points where the matter is discontinuous. It is easy to show that one can make assumptions about the function (]/(2) which lead to the two results we have just stated ( ' ); we will not dwell on this. (') P. DuHEM, Sur la pression électrique et les phénomènes électrocapillaires. i" Partie : De la pression électrique, Chap. I, § II {Annales de l'École Normale supérieure, 3" série, t. V, p. io3; 1888). BOOK V. ELECTRICAL BALANCE AND PERMANENT CURRENTS ON METALLIC CONDUCTORS. CHAPTER ONE. FUNDAMENTAL LAWS OF ELECTRICAL BALANCE ON METALLIC CONDUCTORS. § 1 - Condition of electrical equilibrium. The internal thermodynamic potential of an electrified system is given by the equality (i 5) of the previous chapter (0 ,f = E(r - TS) + w + 0ig',+ e25r2-f-... + e"^". Let us suppose that we have the disposition of the various parts of the system, their physical and chemical state, and let us propose to know according to which law a given electric charge will be distributed on a similar system. Let's imagine that we vary the electrical distribution without moving the various parts of the system; since such a modification does not generate any external work, we can see that we will obtain the condition of electrical equilibrium by writing that, for any modification of this kind, that is (■j.) E8(r - T2)-+-8W -1-0(01 5-1-1-02 5f2 + ...+ e "gr"):= o. Let's suppose that the system is formed by conductors on which we can modify the electrical distribution without modifying the physical and chemical state of any of them, which we will express briefly by saying that it is formed by conductors 368 BOOK V. - METALLIC CONDUCTORS. metal. Then, in any change of electrical distribution, we will be able to suppose invariable the physical and chemical state of the various parts of the system, which will give us 8(r - TS) = o, 801 = O, §02 = 0, ..., O0" = 0, and reduce the equality (2)3 (3) 8W-+- ©1 og-i-i- 02 8g'2-^-- .-HÔraSçr,, = o. Let us first imagine the following modification: The charge ùq passes from the element C) to a point of the element t'2. We will have (4) 8^1= - ogr, oq^ = ^q\ 8^3 = 0, ..., 85-,^ = o. Let's calculate 8W. The initial value of W is L \''12 ''13 n/i/ ?.(?+---?" The final value is L \ ^^12 ''13 ''!" .^■23 "2rt + . , Qn 1n-\n'\n We have therefore 8W = -s^^+sS^^^ + il+...4--il ''12 ''21 ''23 ''in Let V|, Vo be the respective potential levels of the elements CHAP. I. - THE KLECTRK EQUILIBRIUM ^-1 P2- We will have '"12 fl" end '-21 ''23 ^2/1 369 which will give, neglecting the infinitesimals of the second order, (5) 8W = £(V,-V,)S^. By virtue of equalities (4) and (5), equality (3) becomes The particular modification we have just considered thus leads to the following consequence: For V electricity to be in equilibrium on a metallic conductor whose all points are at the same temperature, the quantity (eV-f-B) must have the same value at all points of the conductor. We will now show that this condition is sufficient, i.e., if it is satisfied, equality (3) takes place for any variation in electrical distribution. To prove this, let us first note that, in any modification of the electrical distribution, we have (6) 8W = £(Vi ^^ -H V2 05-2 + - .+ V" 8^"), equality which can be proved as we proved equality (5). Equality (3) then becomes (s V, + e, ) 8^1 -t- (s V2-i- 62) 8^2 -^ . . + (e V" 4- e") Sgr,, :^ o. Suppose that the elements c, ..., vi, form a first conductor C, at any point of which (e V-j- 0) has the same value A ; that the elements i'A+i, - - ., ^-t form a second conductor C, isolated from the first, at any point of which (s V + 0) has the same value A', etc. The previous equality will become A(85ri-f-. . . -{- Zqk) -+- A'(8^A-(-i-f-. . .-H og-/) -H. . - o. D. - I. 24 370 BOOK V. - METALLIC CONDUCTORS. Now, according to the law of conservation of electricity (Liv. I, Chap. 1), we have 8g'i + - - --+- ^qk = 0, the previous equality is therefore proved. Thus the equality (7) £ V -t- 6 = consl. is the necessary and sufficient condition for the electrical equilibrium on a metallic conductor whose all points are at the same temperature. Is it possible to find an electric distribution having at the various points of the conductors a finite solid density, except on certain surfaces at all points of which it would have a finite surface density, and satisfying this condition? This would be impossible if the function 8 were not a continuous function in the whole extent of a given conductor; if, moreover, it were not regular in the whole space occupied by this conductor, except perhaps on certain surfaces. We are thus led to make the first of the hypotheses about the quantity indicated in § 4 of the preceding chapter. If we admit, on the quantity 0, the hypotheses indicated in the paragraph in question, we will see without difficulty that equality (7) leads to the following consequences: 1° Inside a conductor on which the electrical equilibrium is established, there is no electrified surface; 2° At any point inside such a conductor, V electricity has a finite solid density given by V equality (8) ■ p=7^Ae. This density depends only on the constitution of the conductor in the vicinity of the point considered; it varies in a discontinuous way only if this point crosses a surface of discontinuity of the conductive substance. cnAP. I. - Electrical balance. 871 Consider a region of a conductor filled with a homogeneous substance. will then have the same value at any point whose distance from the boundaries of this region is greater than the molecular activity ratio. Equalities (y) and (8) will then become V = const., p = o. It is therefore only in a region formed by a homogeneous substance that the potential function has a constant value and the electric density a zero value. We thus find the fundamental laws of Poisson's theory, but we find them surrounded by restrictions that Poisson's theory could not have predicted. § 2 - Stability of the electrical equilibrium. Is the electrical equilibrium always stable? In other words, does an electrical distribution that satisfies the condition {'j) correspond to a minimum of internal thermodynamic potential? To solve this question, let us look for the sign of 8-J. We have As fine wire r^s V ,j = - - - -1 1- . . . H ; ' fnl rni l'n-in (9) { ■ '^^..3 372 BOOK V. - THE MtTALLIQUE CONDUCTS. So we can easily find the equality l L Vn2 f'is ri,J '2,1 / >'ll-}ll .1 Let us imagine that we place, in the element r, a charge 5^, ; in the element Ç21 wa charge 0^25 - - -, in the element (^", a charge oq,i ; this electric distribution will admit an electrostatic potential ^, and we will see without difficulty that equality (9) can be written Now we have seen [Liv. I, Ghap. IX] that the electrostatic potential of any electrified system is positive; the quantity o^W is therefore positive; so is t'^§^ and, since any equilibrium distribution necessarily makes the internal thermodynamic potential a minimum, the electrical equilibrium on a system of metallic conductors whose points are all at the same temperature is certainly a stable equilibrium. CIIAP. II. - HOMOGENOUS CONDUCT. StS CHAPTER II. ELECTRICAL EQUILIBRIUM ON HOMOGENEOUS CONDUCTORS. LAWS OF ELECTRICAL DISCHARGE. § 1 - Electrical balance on homogeneous conductors. We will not understand here by homogeneous conductor a conductor having the same physical and chemical constitution and the same density at all its points. We shall admit that bodies which appear to us to be homogeneous have substantially the same density at all points whose distance from the terminal surfaces is greater than a quantity [x which is of the order of the radius of molecular activity; but if a point M lies within such a body A at a distance /, less than [jl, from the surface which separates body A from another body B, homogeneous in appearance, we shall admit that the density at point M depends : i" Of the nature of the body A, of its temperature, of the density (|u'it presents far from the terminal surfaces; 2" The nature of the body B, its temperature, the density (|u'it presents far from the terminal surfaces; ?)" From the distance / from point M to the surface that separates the two bodies A and B. These are the assumptions that Poisson introduced in i83o in the theory of capillary action. We will propose to investigate the laws of electrical equilibrium on a system of metallic conductors constituted as we have just indicated. In the present chapter, we will limit ourselves to the case where all the conductors are made of the same metal i and confined to the same insulator o. Let a point M, situated at a distance /, less than()v -f- jx), of the surface which limits a similar conductor. It is easy to see, by referring to the definition of the quantity 0, that will have at this point a value 6 (/), variable with /. The form of the function 374 LlVnE V. - METALLIC CONDUCTORS. lion S(l) depends on the nature of the conductor i and the insulator o, their densities, their temperature. According to equality (8) of the previous chapter, we will have at this point a solid electric density given by the formula (,) p=:^A0(O. According to this equality, p is a function of / which cancels when / becomes greater than ()^+ [x) and which depends on the nature of the conductor (i) and the insulator (o) without depending on the electrical actions to which they are subjected. We thus arrive at the following propositions: When the electrical equilibrium is established on a homogeneous conductor, there is no electricity at the points inside this conductor whose distance to the terminal surfaces exceeds {\ -\- [x). The points whose distance to the end surfaces is less than ().-f-[JL) are electrified. The electrical density is the same at all points on a surface parallel to the end surface. The law according to which Vélectrisation varies, when Von passes from a surface parallel to the terminal surface to another similar surface, depends only on the nature and state of the conductor and insulator away from the terminal surfaces; but not on their shape nor on the particular electrical actions to which the conductor is subjected. Let's take a point M on the surface that separates the conductor from the insulator; at this point, the normal to the surface towards the inside of the conductor has the direction N, and the normal towards the outside has the direction Ne. If V is the potential function, the surface density at point M has the value \ ( d\ àN\ _ y de(o) I dN If we put CHAP. II. - HOMOGENEOUS CONDUCTORS. 375 the density at point M will have the value (4) (T = 2 + A. The surface electric density at a point on the surface of a homogeneous conductor in equilibrium is the sum : i" Of a density S, given by V equality (2), which depends only on the nature that the conductor and the insulator present away from the terminal surfaces, but not on their shape and the particular electrical actions to which the conductor is subjected ; 1° A density, given by the equality (3), which depends on the electrical actions which are exerted in the system and on the shape of the conductors which compose it. We will give to the whole of the layer of density 2 spread in a uniform way on the surface of the conductor and of the electrified layers which follow one another in the interior of this conductor up to a distance (). + {x) from the surface the name to^ natural electricity of the conductor. The electric layer of density A will be Vélectricilé communicated. Let's study the properties of the natural electricity of the conductor. Let us take, on the surface S, an element rfSH-Se(o)i - V)-(-M'(?-V')] and, therefore, E Au= ^[MCC? - V) + M'(9- V')]. Since the exciter has never carried more than light loads, a very small amount of external work was enough to set it in motion. Moreover, its living force is null at the beginning and at the end of the modification. If therefore Q is the quantity of heat released in the modification in question, we will have (20) EQ= ^[M(V-\'^) + M'(V'--(:;)]. It is the same as the amount of heat that would be given by Glausius' theorem, if we suppose that the system had remained motionless and that the electricity had passed directly from one of the conductors A, A' to the other. Second remark. - It is assumed that one of the two conductors is formed by a solid part A (Jig-. 72), connected by an extremely fm wire F, to a ball of small dimensions B. One end of the exciter is placed in contact with the conductor A', and the other with the ball B. The experiment 384 BOOK V. - THE METAL CONDLCTELKS. shows that the heat released by the decliff is almost exclusively given up to the medium surrounding the wire F. This fact is related to Joule's law, which we will study in a later chapter. Fig. 7:^. It allows to determine experimentally the amount of heat released in the discharge. 11 It is sufficient to use a kind of air calorimeter Fig. 73. invented by Snow Harris in 1837 (' ). The wire F {fig- 73) is enclosed in a glass envelope terminated by a manometer at ( ' ) Snow Harris, On the relative powers of varions metallic substances as conductors of electricity { Philosophical Transactions, t. CLVIII, p. 18; 1827). CHAP. H. - HOMOGENEOUS CONDUCTORS. 385 free air. At the moment of discharge, the wire, initially heated, quickly reaches a temperature equilibrium with the air contained in the glass envelope. If we denote by Tq the initial temperature of the wire and of the air mass; by T, their final temperature; by m the reduced weight in water of the wire and of the air mass, the quantity of heat released has the value ci(T, - Tq). Thanks to the presence of the manometer, the apparatus forms an air thermometer with an approximately constant volume. If we denote by (P, - P^) the variation of pressure, we will have P.-Po=aPo(T,-To), a being the coefficient of expansion of the air; we will thus have The quantity m remaining constant in the various experiments, this equality will make us know numbers proportional to the quantities of heat released in these experiments. Riess ('), who perfected the Snow Harris manometer, used it to determine, under various circumstances, the amount of heat released by the electric discharge. Clausius (2) compared the results of Riess' experiments with the consequences of the theory. We shall indicate here some of these experiments. § 4 - Complete discharge of a capacitor. Experiments of Riess. A capacitor is charged; the inner armature has received an amount of electricity a; the outer armature carries a communicated charge - b. The potential level of the communicated distribution is V on the first conductor and o on the second. Moreover (Liv. II, Chap. XI, § 2) b = ma, m being the first Gaugain coefficient. Let's put the two armatures in communication with each other; (') Riess, Lehre der Reibungs-Elektricitàt; Berlin, i863, (') Clausius, loc. cit. D. - I. 25 386 BOOK V. - METALLIC CONDUCTORS. a discharge will occur giving off a quantity of heat Q which, according to formula (20), is given by the equality EQ= i[aV + (a - 6)t;;] = ia[V + (i_m)t;;]. In particular, let's assume that the outer frame completely envelops the inner frame. Then, according to the principles explained in Book TI, Chapter V, we will have a = 6, m =1 and therefore (21) EQ==îaV. Let us imagine a battery formed of n identical bottles, arranged in any manner, each having its internal armature in communication with the source and its external armature on the ground. According to the principles laid down in Book III, Chapter V, each of these bottles exerts a null electrostatic action on the outside; consequently, each of them will electrify as if it existed alone. If we discharge the battery, the amount of heat released will be Moreover, we can write "= £GV, C being a constant (capacity of the cylinder) which depends on the construction of the cylinder. We will thus have, by designating by A the total load, equal to /la, of the internal armatures, "I t A2 (-) Q=^G7r The amount of heat released by the discharge of a battery of identical jars is proportional to the square of the total load taken by the inner fittings, and inversely proportional to the number of jars. Riess verified this law by the following experiments. He has first of all CH.VP. II. - HOMOGENEOUS CONDUCTORS. 887 On board took a battery of five jars, to which he communicated variable charges; he then compared the number D of divisions by which the liquid of the thermometer rose, as a result of the different discharges, with the number D' by which it should rise according to the formula (22); he found the following results : A. D (observed). D' (calculated). 3 1,5 1 ,6 4 3,0 2,8 5 4,5 4,4 6 6,5 6,3 7 8,8 8,6 8 11,3 11,3 9 i4,3 14,3 10 16,7 17,6 11 then varied the number n of bottles while keeping the internal armature of the battery at a constant charge A, and found the following results: n. D (observed). D' (calculated). 2 i3,4 i5,8 3 9,7 10,6 4 7,3 7,9 5 6,5 6,3 6 5,5 5,5 We see that the agreement of formula (22) with experience is, in general, very satisfactory, and of such a nature as to leave no doubt as to the accuracy of the law we have just stated. We5 have assumed only, so far, that the outer frame of the Lejde bottle completely enveloped the inner frame. Let us now assume, as we have already done (Liv. ni, Ghap. IV, § 4), that the inner surface of the outer frame is a level surface of the inner frame. In this case, the charge a of the internal armature has the value [quote, equation (17)]. V being the potential level to which the internal armature would be brought if we distributed on this armature, taken in isolation, an electric charge equal to the unit; u being the value that would have the function 388 BOOK V. - METALLIC CONDUCTORS. of this distribution at any point on the inner face of the outer frame. Let us suppose that the two armatures are very close; let M be a point of the internal armature; let N^ be the normal at this point to the internal armature directed towards the outside of this conductor; let a be the density that an electric charge equal to the unit distributed on the internal armature takes at this point; let S be the very small distance between the two armatures; let dS be an element, drawn around the point M, of the surface S of the internal armature. We will have the various equalities that give I dv a - "4^ dN, S' T^S: = I, di^ u - V at I l c dS Let A V be the average thickness of the insulating layer between the two reinforcements, given by the equality Ë - C - The previous equality will become 1 Ë 471 Â' Equality (28) will then become I S [\-K A and equality (21), (24) EQ = 2Tr£A^'. Ley de bottles, formed as we have just indicated, in which the two armatures are at the same very small average distance^ give, whatever their shape, when they are discharged, a quantity of electricity proportional to the square of the charge of V internal armature, and in inverse proportion to the surface of this armature. CHAP. II. - HOMOGENEOUS CONDUCTORS. 889 This law was found experimentally by Riess ( * )- M. Helmholtz (^) and Clausius (^) established it theoretically. The previous demonstration is due to M. J. Moutier (^). § 5 - Discharge of a capacitor by successive sparks. A capacitor can be discharged by successive sparks: it is enough to isolate the capacitor and to put in communication with the ground, first the internal armature, then the external armature, then again the internal armature, and so on. This discharge did not give rise to measurement experiments; the theory is very simple; it was given by Mr. J. Moutier (5). Let's look at the amounts of electricity that flow into the ground with each spark. Let A be the inner frame and B the outer frame. Initially, conductor A has an electric charge a and conductor B an electric charge - b. When we put conductor A on the ground, with conductor B remaining insulated, we carry out Gaugain's third experiment (Liv. II, Chap. III, § II). After this first discharge, conductor A remains charged with the quantity at= mm! a of positive electricity. The first spark therefore brings to the ground a quantity <7i = (i - mm')a of positive electricity. When we then isolate the conductor A, carrier of the charge a, = mm' a, and we put the conductor B in communication with the ground, we carry out the second experiment of Gaugain, but assuming j the charge of the body A reduced in (') Riess, Ueber die Erwàrmung in Schliessungsbogen der electrisclien Batterie {Poggendorff's Annalen, Bd. XLIII, p. 47; i838). Lehre der ReibungsElektricitât; Berlin, i853. (") H. Helmholtz, Ueber die Erhaltung der Kraft, p. f\Z (Berlin, 1847. - Helmholtz wissenschaftliche Abhandlungen, l. I, p. l\b). (' ) R. Clausius, On the mechanical equivalent of an electric discharge and the heating it produces in the conducting wire ( Mechanical theory of heat. Translation Folie, t. II, p. 60). (♦) J. Moutier, Cours de Physique, t. I, p. 499; Paris, i883. (') J. Moutier, Cours de Physique, t. I, p. 4^9 and p. 486; Paris, i883. 3jO BOOK V. - METALLIC CONDUCTORS. the ratio mm! A similar reduction must be made in the charge of the conductor B, which thus contains, after the second spark, a negative charge bi = mm'b. The second spark therefore brings to the ground a quantity of electricity ^2 = m(i - mm') a of negative electricity. At the moment of producing the third spark, we are in a state similar to the initial state, but where all the charges have been reduced in the ratio mm' . The third spark therefore brings to the ground a quantity of positive electricity ^3= mm' {i - mm') a. We will see that the spark of order (2/^-1-1) will bring to the ground a positive quantity of electricity qin+\ - m"^m'"(i - mm') a, while the spark of order (2/^ + 2) takes to the ground a quantity of negative electricity Çin+i = m"^+^ m''^(i - mm' )a. Let's look for the amount of heat that each of these sparks would give off in the Riess thermometer. Before the first spark, the electrostatic potential has the value Wo- -aY. 2 After the first spark, the conductor B is at potential level \'; the electrostatic potential has the value W,, and we have Wi = --6V'. 2 The Gaussian identity, applied to the initial and final states, gives, noting that ", is the final charge of conductor A, aiY = - bY'. Thus we have Wo- Wi= -(i - m/n')aW. CHAP. II. - HOMOGENEOUS CONDUCTORS. 89 1 The amount of heat released by the first spark is value Q,= -^(i - mm')a\. To find the amount of heat released in the second spark, we note that before this spark, conductor A is at potential level o and carries a charge "i; conductor B is at potential level V and carries a charge - b. The electrostatic potential has the value W, = --6V'. 2 After the discharge, conductor A carries charge a, and is at potential level V". Conductor B carries a charge - b^ and is at potential level o. The quantity ^, is given by bi - mm' ù. The electrostatic potential has the value W2=-aiV". Moreover, the Gaussian identity gives So we have and W2 = --6,V' 2 W,-W2 = - J(^'-^',)V' = - m7n'{i - m.m')aV. The amount of heat released by the second spark has the value Q2 = - =; mm' ( I - mm' ) a V = mw' Qi . 2ll> We would also find that the amount of heat released by the third spark has the value Q3=m2m'2Q,. 392 BOOK V. - METALLIC CONDUCTORS. The quantities of heat released by the successive sparks decrease in geometric progression of reason mm' . §6. Cascaded batteries. Let us suppose that we have {n-\-} bottles, identical to each other, and far enough apart from each other to have no influence on each other. The internal armor Ao of the Soil The first {Jig- 74) 6st is connected to the source at the potential level Vo- The external armature Bo is connected to the armature A of the second, and so on. The ar CHAP. II. - HOMOGENEOUS CONDUCTORS. SgS The last one is externally mature on the ground. We have thus a battery charged in cascade. The theory of cascade loading was the subject of research by Green ('), Clausius and Béer. M. J. Moutier has given this theory a very elegant form (-), in the case where the two surfaces S', S" which limit the external framework of each bottle are level surfaces of the internal framework S. Let Vo, V,, V2, ..., V" be the potential levels of the armatures Ao, A,, A2, ..., A". Let's consider first the last bottle. It carries no electric charge on the surface S"; on the surface S' it carries a charge - a,i and on the surface S a charge On- ' An electric charge equal to unity, in equilibrium on the surface S, would bring this surface to the potential level v; a charge equal to unity, distributed on the surface 2', would bring any point inside this surface to the potential level v' (^); finally a charge equal to unity, distributed on the surface S", would bring any point inside the surface 2" to the potential level v" . With these notations, we can already write For the [n - iy<'"" bottle, the inner armature carries a charge a,i-\; the surface 2' a charge - ""-4, and as the two bodies B"_i and A" form an insulated conductor carrying a total charge of zero, the surface 2" carries a charge (""_( - ""). The principles laid down in Liv. III, Chap. IV, easily give a"-iv - a"_i (>'-+- (a"-i - a")t>" = V"_i. We will thus arrive from one step to another to have the two series (') Green, An essay of the application of mathematical analysis to the theories of Electricity and Magnetism, Art. 8 (Nottingham, 1828, Green's mathematical Papers, p. 47 )- (') J. Moutier, Cours de Physique, t. I, p. 491; Paris, i883. (' ) To make the formulas we are about to write coincide with those obtained in Book III, Chap. IV, we would have to pose v' = u, v" = u'. 394 BOOK V. - METALLIC CONDUCTORS of equations ttnV - anV' =Yn, aa-i{v - v'-\-v") - a,iv" = V,,-, cin-iiv - v'-\- v") - a,i-\ v"= V"_2, (25) a2{v - v'-{-v") - a^v" =¥2, ai{v - v'-\-v") - anv" = Vi, a^iv - v' -^ v") - a^v" = Vq. (a"_i - a")p" =^V", (a,j_2 - ""~i)<^" = V"_i, (26) ("1 - a<))v" = V2, ("0-"i)^'"= Vf 1 "1 1 "1 -F- "3 = v" V ~ - t"' -r - IV V - t^'-r ■IV"' Between the equalities (aS) and (26), we can eliminate the potential functions; we then have i( ao - - ai ) p" = "1 ( c - v' -4- p" ) - "2 ^'''j (ai - "2)*^"= "2(^' - c'h-p") -- "3"^", or (28) The charges of the internal armatures of the bottles are subject to the following law: If V we take three consecutive bottles, the sum of the charges of the internal armatures of the two extreme bottles is in a constant ratio to the charge of V internal armature of the average bottle. The equations (aS) give Vo-f- V2 = (ao-H a2)((^ - t^'-i- v") - {a^t-^ a^)v\ or, according to the equalities (28), aip"(p - v' -\- v") a^v"'^ V0+V2 = [ai(v - v'-hv") - a^v"] V - V -{- IV (î9) CIIAP. II. - HOMOGENEOUS CONDUCTORS. SgS According to the equalities (20), this one can be written V, The potential levels of the internal armatures thus follow the same law as the charges. Let us add member by member the last of the equations (aS) and the equations (27). We will find (3o) \q = (uo-h ai-h. . .+ an)iv -v'). This equality, compared to equality (aS), highlights the following theorem: The sum of the charges of the internal armatures of a cascaded battery is equal to the charge that the internal armature of the first cylinder would take if it were charged alone. (^this beautiful theorem is due to Green. Let's suppose that we leave the external armature of the last cylinder on the ground and that Ton puts the internal armature of the first cylinder in communication with the ground. The battery will be brought back to the neutral state. If W is the initial electrostatic potential of the system, we will have, for expression of the heat released by this discharge, EQ = W. However, the general formula w=i2. gives easily W= -aoVo 2 The heat released has therefore the value (3i) Q:^-^aoVo. If we had charged only the first cylinder, and discharged it as we have just done for the battery, the discharge would have, according to equalities (21), (23) and (3o), released a 396 BOOK V. - METALLIC CONDUCTORS. amount of heat By comparing this equality with equality (3i), one sees that the heat released by the complete discharge of a battery charged in cascade is all the smaller as the number of the bottles is larger. In the particular case where the two surfaces S', S" differ very little, as it happens if the external armature is formed by a simple metal sheet, we have approximately We then easily find the equalities (3-2) (33) If the external armatures are very thin, the loads of the internal armatures and their potential levels decrease in geometric progression. Béer (^) erroneously stated that, in this case, the charges of the internal armatures were all equal to each other and that their potential levels decreased in arithmetic progression. In this case, the amount of heat released by the complete discharge of the battery has the value Q = -^. - - - - "^ "0 CTi ""-1 V "1 "2 a,i v' Vi v^--' V v' A = ao -T- "1 -I- . 4- a". Let's put We will have (34) A= -, an. We can therefore write, by designating by Q' the quantity of heat that would be released by the complete discharge of a cylinder C) Béer, Einleitung in die Elektrostatik ..., p. 102 (Brunswick, i865). CHAP. II. - HOMOGENEOUS CONDUCTORS. 897 unique, (35) V ' ^^ Q V - 1>' v"^ In the particular case where the two armatures are very close to each other, we can easily find that this equality becomes Q'=(n + i)Q. If Von forms a cascade battery by means of bottles whose outer frame is very thin and whose two frames are very close together, the amount of heat released by the complete discharge is inversely related to the number of bottles. This law was found experimentally by Riess [loc. cit.] and theoretically by Glausius [loc. cit.]. 398 BOOK V. - METALLIC CONDUCTORS. CHAPTER m. THE INTENSITY OF THE CURRENTS. § 1 - Currents flowing in the mass of a conductor. If the state of a conductor at time t were entirely determined by the electrical distribution it carries at time t^ the properties of this body would have to remain the same, either if this electrical distribution remained after time t what it is at time ?, or if it underwent changes after time t. But this is not the case. The conductor on which the distribution varies will exert certain actions on a magnet that a conductor on which the distribution is invariable does not exert, whatever this distribution may be. It is therefore necessary, in order to define an electrified system, to use a more complicated representation than the one we have used up to now; to add new variables to the variables which determine the electrical distribution, these last ones disappearing in the particular case where, on the conductor, the distribution remains independent of time. The definition of these new variables is mainly due to G. -S. Ohm (' ), Smaasen (2) and G. Kirchhoff (3). (' ) G.-S. Ohm, Die galvanische Kette, mathematisch behandelt {Berlin, 1827. - Translated into French by Gaugain; Paris, 1860). (") Smaasen, Vom dynamischen Gleichgewicht der Elektricitàt in einer Ebene oder in einem Kôrper {Pogg. Annalen, Bd. LXIX, p. i6r; 1846). - Vom dynamischen Gleichgewicht der Electricitàt in einem Kôrper und in unbegrânztem Raume{Pogg. Annal, Bd. LXXII, p. /|35; 1847). (^) G. KiRCHHOFF, Ueber den Durchgang eines élektrischen Stromes durch eine Ebene, insbesondere durch eine kreisformige ( Pogg. Ann. Bd. LXIV, p. 497; 1845. - Kirchhoff's Abhandlungen, p. i). - Nachtrag zu dem vorigen Auf CHAP. III. - the intensity of the currents. 899 These ph3'sicians were led to the definition of the variables in question by the comparison of the motion of electricity with the motion of fluids; a comparison of this kind had already led Fourier to the definition of the principal quantities which appear in the theory of the propagation of heat. Let M be a point inside a conductor. We will suppose that this point M corresponds to a geometrical quantity F which we will name \electric flux at point M at time t. This quantity is related to the variation of the electric distribution inside the conductor by the following convention: Around the point M {fig. 70), let us trace in the mass of the con Fig. 7,5. Let AB = i/w be an element. Let N be the normal to this element in a given direction. Let F be the flux at the point M. The variation undergone by the electric distribution on the conductor in time dt is the same as if, during this time, the element doi had been crossed, in the direction of the normal N, by a quantity of positive electricity dq, given in magnitude and in sign by ([) dq = Fcos(F,N)diodt. This equality (i) can be written in a slightly different way. satze {Pogg. Ann. Bd LXVII, p. 344; i846. - A'. Abhandl., p. 17). - Ueber die Aujlôsung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Strômen gefiihrt wird {Pogg. Ann, Bd. LXXII, p. 497; 1847. - A". Abhandl., p. 22). - Ueber die Anwendbarkeit der Formeln fur die Intensitàten der galvanischen Strômen in eineni Système linearer Leiter auf Système, die zum Theil aus nicht linearen Leitern bestehen {Pogg. Ann, Bd. LXXV, p. 189; 1848. - K. Abhandl., p. 33). - Ueber eine Ableitung der Ohm'schen Gesetze, welche sich an die Théorie der Elektrostatik anschliesst {Pogg. Ann. Bd. LXXVIII, p. 5o6; 1849. - K. Abhandl., p. ^9). - Ueber die stationàren elektrischen Stromungen in einer gekriimmten leitenden Fldche {Monatsber. der Akad. der Wissenschaften zu Berlin, 19 July 1875. - K. Abhandl., p. .56). 400 BOOK V. - METALLIC CONDUCTORS. Let us take three axes with rectangular coordinates; let m, ç, w be the components of the flow F along these three axes. We can write (2) dq ~ [acos(N, x) -t- v cos(N,j) -t- w cos(N, z)] dia dt. This equality (2) will lead us to a formula that will clearly highlight the relationship between the (lux and the change in electrical distribution. Let us draw, inside the conductor, a closed surface S. Let M be a point of this surface and N,- the normal to this surface at the point M; this normal is directed towards the interior of the space limited by the surface S. The change of electric distribution on the conductor is the same as if each of the elements ûfS of the surface S let, in the time dt^ penetrate inside this surface a quantity of electricity dq = Yu cos(N/, x) -\- V cos(N/,j') -h v cos( Nj, z)]"iS dt. The change in electrical distribution on the conductor must therefore have the effect, during the time dt., of increasing by dt V [m cos(N/, x)-^ V cos(N/, J-) -+- "'cos(N,, z)dS the amount of positive electricity contained in the surface S. But, on the other hand, if we denote by p the electric density at the "point" (x, y, z) at time t, a density that we assume to be finite at any point inside the surface S, the change of electric distribution during time dt will increase the total electric charge enclosed in the surface S by dt I I i Yfdx dy dz, the Integration extending to all the space inside the surface S. We must therefore have, whatever the shape of the surface S, V [acos(Nt-,a7)-f-(^cos(N;,jK)-H"^cos(I\;,^)]â?S - / / j-dx dy dz =^ 0. This equality can be transformed. Suppose that the quantities^ M, V., w are continuous as well as their partial derivatives of the first order in all the space enclosed by the surface S; that it CHAP. lit. - the intensity of the currents. 4oi due be the same for p and -r^". The previous equality will become Under the hypotheses made, it cannot take place for any surface S, unless we have, at any point where the indicated conditions are verified, (3) of dx at dw ùt This equality does not apply to theK various points of a surface along which the quantities m, p, w, p may be discontinuous. Let us consider this case, assuming that this surface can carry a variable surface electrification of density n. A similar surface S {fig- 76) separates two regions i and -i Fig. 76. / / ■§ J x^ - \ of the conductor. On this surface let us take an area and, by the contour AB of this area, lead straight lines normal to the surface S. We limit these lines by two surfaces Si, S2, parallel to the surface S, situated one in the region i, the other in the region 2, both infinitely close to the surface S. Let A, B,, AoB^ be the areas cut on these two surfaces by the considered ruled surface. To the nearest terms of the order of A,Ao, the amount of positive electricity that enters the closed surface A/BiAoBo during the time dt can be written - c?/ X [ Wi cos(Ni, a?) -+- t^i cos(Ni, y) -t- (V| cos(N,, 3) KJxn D. - I. Ui cos(N2, a:)H- Vi cos(N2, j) -f- w^ cos(ÎN2, 3)] rfS 26 402 BOOK V. - METALLIC CONDUCTORS. and also dt^ $dS. dt By equating these two quantities, we can easily see that we must have, at any point of the discontinuity surface S, / i(iCos(Ni, x) ■-!- i'i cos(Ni, j) -H Wi cos(N], z) ^^' I + "2 cos(N2, a?) -i- P2 cos(N2,jk) -I- M^a cos(N2, -z) = -7- - Equalities (3) and (4) show us to what extent the fluxes on the one hand and the electric densities on the other hand can be regarded as independent variables. One can always, for a particular value of t, arbitrarily give oneself the magnitude and direction of the electric flux at each point of the conductor, and the magnitude of the solid or surface density of electricity at each point of the conductor. But, for later values of time t, it is no longer permissible to give oneself arbitrarily anything other than the magnitude and direction of the electric flux at each point of the conductor; for the densities, both solid and superficial, are then determined, for all values of t, by the equalities (3) and (4). At the surface that separates the conductor from the insulating medium that surrounds it, we have, according to equality (4), (5) u cos(N/, x) -h V cos{Ni,y) -+- w cos(N/, -z) = - -r- Some authors have admitted that we always have, in this case, u cos(N;, x) -^ V cos(i\;,^) -\- w cos(Nj, ^) - o. - But then no current could vary the electrical distribution on the surface of a conductor; it could never change, which is inadmissible. § 2 - Uniform currents. If the electric flux is zero at any point of a conductive body, we say that V electrical equilibrium is established on this conductive body. According to the equations (3), (4) and (5), the solid electrical density CHAP. III. - the intensity of the currents. 4^3 OR surface then maintains a value independent of time at any point of the conductor. But the solid or surface electric density can maintain, at any point of the conductor, a value independent of time without the electric flux being zero at any point. Indeed, it is enough for the currents not to cause any variation of distribution on the surface of the conductor or in its interior, that we have at any time, for any point inside the conductor, -from ()(' ôiv dx dy dz ' for any point on the surface separating the conductor from the insulator, (7) u cos(N,-, x) -h V cos(N/,^) -r- w cos(N/. z) - o; finally, for any point of a discontinuity surface of the conductor, ( Ml cos(Ni, ip) -H (^1 cos(Ni,_7) -4- (iPi cos(Ni, ^) ( -4-"2 cos(N2, x) -T- t'2 cos(N2,_y) -f- "'2 cos(N2, z) = o. Such a current, which at each instant brings as much electricity to any point inside the conductor or on its surface as it carries away, is said to be a uniform current. When at any point of a conductor the electric flow is independent of time, the current is said to be constant. A current that is both uniform and constant is called permanent. § 3 - Linear currents. Suppose that an infinitesimally small plane area A {Jig. 77), will Kig. 77. (8) It moves in such a way as to remain constantly normal to a line LL'. It sweeps a volume which we will suppose filled with conductive matter and which we will name 4o4 BOOK V. - METALLIC CONDUCTORS. Let's take a wire. If this wire is traversed by any electrical flux, we will say that it is traversed by a linear current. Let N be the normal to the area A, on a given side of this area. It is at the same time the tangent at L to the curve LL'. The area A is crossed, in the direction indicated by the normal N, during the time dt, by a quantity of electricity c/Q = ^^C [mcos(N, 37)4- pcos(N,7)-+-"'cos(N,3)]£/A. If we put (9) J = V [a cos(N, x) -\- V cos(N,^) -\~ w cos(N, z)\dk, we will have ^Q = J dt. We then say that J is Vintensity at point L of the linear current. Let us assume, first of all, that the wire LL' has no discontinuity surface between the points L and L'. It is easy to see that, under these conditions, the intensity J is continuous between the two points L, L'. Between two sections of the conductor, comprising between them a length ds of the curve LL', there accumulates during the time dt a total quantity of electricity z- as dt. ds Let us now assume that the wire has a surface of discontinuity, and, in order not to complicate our reasoning too much, let us imagine that this surface coincides with a normal section A. This section divides the wire into two regions i and 2. When we go from the first to the second, the intensity changes abruptly from the value Ji to the value Jo. In the time dt, a quantity of electricity accumulates on the surface A (Ji -i2)dt. If S is the average surface density over the area A, we have (.0, g= ■(,,_,,). CHAP. III. - the intensity of the currents. 4o5 Let us consider, on the other hand, a portion LL' of the wire, along which there is no surface of discontinuity. Let J, J' be the values of the intensity at the origin and at the end of this segment, in time dt, this segment acquires a quantity of electricity {i - y)dt. Let A be the average cross-section of the LL segment; G the average perimeter of this cross-section; a- the average surface electric density; p the average solid density. We will have If we admit that the densities p and o- are, in general, of the same order of magnitude, A being negligible before G, we will have (-0 Let us compare the equalities (lo) and (i i). Since A is negligible in front of G, if we want the electric densities S and ds = ^ ^' doi, \ AUA'B' and the previous equality can be written (2) ^Q = [X') cos(rt, x) -+- \'> cos(rt, y )\ ds dl. The quantities "O and <>, defined by the equalities (i), are what we will call the components of the surface flux at the point As On the upper side of the plate, draw a small rectangle Fig. 83. y/ / ^ D 7 / a//'~" -s^-\ -JD' --■^i' ABCD i^fig. 83), whose sides AB ;= dx and AC ^=-dy are pa CHAP. V. - MOVEMENT OF ELECTRICITY IN A PLATE. 4^1 parallel to the axes Ox, Oy. By the contour of this small rectangle, let us lead normals to the plate, which cut out a parallelepiped ABCDA'B'C'D'. This small parallelepiped contains a total quantity ^ of electricity spread inside it, or on its faces ABCD, A'B'C'D'. We will pose and we will name P the electric density at a point A of the plate. It is easy to see that we have (3) Oy ~ôi' This equality shows that, if the currents are uniform, we must have (1) Ox Oy Let ABA'B' be a rectangular element between two generatrices AA', BB' of the plate edge {Jig- 84)- Let Ng be the normal to this element towards the outside of the plate. Let ds be the length AB. Let ^ds be the quantity of electricity it contains. We can easily find (5) X) coë (rie, ^) -+■ "->? cos(ne, y) = dt If the currents are uniform, the second member will be zero, and we will have, by designating by /?/ the direction opposite to ne-, (6) XD cos(",-, x) -+- V cos(n,-, y) - o. The surface Jlux lines must be tangent to the edge of the plate. 4'22 BOOK V. - METALLIC CONDUCTORS. Let us assume the homogeneous plate and the permanent currents. Let àl be the specific resistance of the substance which forms the plate. Let V be the potential function. We will have, according to Ohm's law, __ ^ dY __ ^ ^^ " ~ ~ ^ 5^ ' ^ ~ ~ Jl 6"/ ■ Equalities (i) will then become AU A' 15' ^■^■"=-iS dy A 1!A' B' d(M. d\ d\ . But the quantities -^j -- - vary in a continuous way, not ^ ox ôy not only inside the plate, but also at the crossing of its faces. Each of these quantities will have, at any point of the element ABA'B', approximately the same value as at point A. We can therefore replace the previous equalities by - ) - denoting here the values that these quantities take at point A. - The quantity (7) ^-f is what we will call the specific resistance of the plate at point A; the introduction of this quantity allows us to write Let us now assume that the plate has a constant thickness, so that the quantity Itt has the same value at every point. Equality (4) will become it must take place at any point of the plate, while at any point of the I CIIAP. V. - MOVEMENT OF THE ELEOTRICIAN IN A PLATE. 4^3 point of the edge of the plate we must have, according to the equality (()), Let us therefore consider a plate with L as its edge (Fig. 85). On this plate, two areas, one bounded by the line Lo, the other bounded by the line L,, are maintained respectively at the potential levels Vo and V,. To determine the perma The first step is to find a function V which verifies the partial differential equation (9) at any point of the area between the lines L, Lo, L(, taking the value Vq at any point of the line Lo> 1^ value V, at any point of the line L, and satisfying the condition (10) at any point of the line L. The components of the flux will then be determined at each point by the equalities (8). The problem thus posed is in the same form as the problem of the permanent motion of the eleclricle in a three-dimensional conductor. But we have methods to solve it which have no equivalent in the case of three variables. These methods, which we cannot detail here (*), are essentially based on a few theorems that we will indicate. Consider a complex variable z - X -^^ iy. Let Z = F(a7 4- iy) any analytic function of this variable. This function (' ) See, for the exposition of these methods, G. Kirchhoff, Voiiesungen iibei mathematische Pliysik-Mechanik, Lesson XXI (Leipzig, 1877). 424 BOOK V. - METALLIC CONDUCTORS. lion can always be put in the form of Z = X + t Y, X and Y being two real analytic functions of x and y. The first is the real part of Z; we will denote it by IHZ. The second is the coefficient of the imaginary part of Z. We will often denote it by Jf Z. The quantity Z should depend on ^ and jK only })ar (jr H- /j-), we will have iTL _ dZ iïL dx i ôy 0: equality which becomes .dY _ .dX dx dy ^ <^y or dX dY dx ^ ¥' dY dY '^ dx From this, we can easily deduce (>i) l d'-X \ dx-^ \ d^Y \ dx* d'-X dy'd'-Y àXdYdXdY_ dx dx ày dy The equalities (i i) show (jue, if we denote by F(^ + iy) any function of the imaginary variable [x 4- iy), the expressions (i3) ^ ^%V{x^iy), (i3 bis) V = J V{x-\-iy) represent two integrals of V partial differential equation ^9^ 'dlc^^-^-dy^ =^ Equality (12) shows that both equations ( îl Y {x + iv) = const, (i4) \ \ Ji ( J Y{x H- ly) = const. CHAP. V. - MOVEMENT OF ELECTRICITY IN A PLATE. 4^5 represent two families of lines that intersect orthogonally. If one of these families represents equipolential lines, V other represents dejlux lines and vice versa. These beautiful propositions show that all the important theorems of the theory of functions of imaginary *'ariables will find an image in the study of the permanent motion of electricity in a plate ('). Any function of the complex variable {x - "jk), conjugate of the previous one, will have analogous properties. We can prove the following proposition: Let F(x -'r iv), G{x - if) be any two analytic functions of two complex conjugate variables; the symbol tl F( jr - t>) -4- -" G(^ - i" represents V general integral of V equation The relations of the theory of functions of imaginary variables with the theory of conformal representation lead to the following theorem, which we will only state: Given two plates P and P', of which V one is the conformal representation of Vautre, which means that any point on one can be matched to any point on the other so that : 1° Any point of the contour of the plate P corresponds to a point of the contour of the plate P' and vice versa; 2" Any point on the electrode contour of plate P corresponds to a point on the electrode contour of plate P' and vice versa; 3° If a point M of the plate P tends towards a point m of the contour of the plate, the point M', which corresponds to M on the plate P', tends uniformly towards the point /?/ which corresponds to m on the <;ontour of the plate P' ; (' ) See V. Klein, Ueber Hiemann's Théorie der algebraischen Funktionen und ihrer Intégrale (Leipzig, 1882). 426 BOOK V. - METALLIC CONDUCTORS. 4" Any infinitely small figure drawn on plate P corresponds to a similar figure drawn on plate P' and vice versa;' If we know how to find the permanent motion of electricity in plate P, we know how to find it in plate P' and vice versa. The equipotential lines and the lines of jlux of one of the plates have respectively for representation the equipotential lines and the lines of Jlux of the other plate. Let us apply the first of these theorems to an interesting special case. The two electrodes are reduced to two points: one M, {fig. 86), Fig. m. of coordinates ( <, ^,), the other Mo, of coordinates {a->, b^)- The potential function at these two points will not be given, nor will it be required to have a finite value. We will only give the intensity J of the current brought by the first electrode and carried by the second. Let's put Cl = "1 -1- ibi, C2 = "2-1- '^25 Z ^^ X -+- ij, "^ z - c^ Let's find the real part and the coefficient of / in the imaginary part of Z. If we put oc - "1 = /'i COSGi, 37 - "2 = r^ COS62, y - ^1 = /'i sin6i, y - b^=^ r it is necessary and sufficient that we have (18) - -f = consl., in any way that the two points M and M' are located on the plate. Here is how G. Ivirchhoff verified the accuracy of this relationship: In the wire of the galvanometer was inserted a thermoelectric cell, maintaining a constant potential difference between the two ends of the wire. In order for the galvanometer to show no current, it was necessary, as shown by the study of thermoelectric currents, that the two ends of the wire touch two points M, M' of the plate, whose potential level difference is equal and of opposite sign to the one that the thermoelectric cell maintains between the two ends of the wire. For two pairs of such points, whatever their position on the plates, the equality (i8) must be verified. G. Kirchhoff first studied pairs of points located on the line M" Mo. We had, in this case, n + ^2 = r\ -H r; = Ml M, = iij, relation which, joined to equality (i8), allowed to calculate /', when /"j was known. G. Kirchhoff compared the values of r\ thus calculated with the observed values of r\, and he found the following results: /*i 5 10 i5 20 25 3o /-', obs 10.4 17.3 22.8 28 3i.5 34.4 r\ obs - r\ wedge -i-o,4 - o,[ - 0,4 -+-0,2 0,0 - 0,2 G. Kirchhoff then studied the couples located on a circle of 43o BOOK V. - METALLIC CONDUCTORS. 5 inches of radius, passing through the two points M', M2, whose distance was, in this experiment, i inch. He found: 10 20 3o 40 5o 60 70 80 25,4 48,3 62,5 70,9 78,7 84 88,75 9^ i- 0,2 -+- 0,3 - 0,4 - i,i 0,0 - 0,3 0,6 0,0 /-l r\ obs r\ obs. - r\ cale. . . These experiments, as we can see, provide a very complete and precise verification of Ohm's law. The problem of the permanent motion of electricity in a plate can be solved for a large number of cases (*). Mr. Quincke and Mr. Adams have given a large number of experimental verifications of the theoretical results obtained (^). Ohm's law is, by these verifications, placed beyond all dispute. § 2 - Currents in a curved blade. We can study the permanent motion of electricity in a conductor bounded by two very close curved surfaces, as we studied the motion of electricity in a plate. Let us consider the upper face of the plate and, on its surface, let us draw a system of orthogonal curvilinear coordinates. This system is formed by two families of lines: the lines [3, that represents the equation a = const, and the a-lines, as represented by the equation P = const. If ds denotes the distance from point (a, (3) to point (a + doc, p -+- d'^), we have (19) ds^-=zA'idoc'^-{-B^dp, A and B being two positive functions of a, [3. (') G. KiRCHHOFF, Vorlesungen iiber niathematische Physik-Mechanik , XXI" Lesson (Leipzig, 1877). - E.Mathieu, Théorie de l'Électrodynamique, Chap. IV et V. Paris, 1888. (") Quincke, Ueber die Verbreitung eines electrischen Stromes in Metall-platten {Poggendorff's Annalen, t. XCVII, p. 882; i856) ^ Adams, Proceedings of the royal Society of London, Bakerian Lecture, t. XXIV, p. i; 1875. CIIAP. V. - MOVEMENT OF ELECTRICITY IN A PLATE. 43l The small rectangle, whose vertices are the points (a, P), (a + ^/a,P), (a, ? + rfp), {% + doL, ^ -^ d^ has area (io) diù^ hBd%d^. The surface fluxes, in a similar blade, are defined as in a plate. At the point (a, ^), we will denote by/ the component of the flow along the line a, and by "- the component of the flow along the line ^. Let P be the average density of electricity at the point (a, ^) of the plate. We can easily find that (it) AB -,- = - ( - ^ -f- -,"- If S is the average density at the edge of the plate and n/ is the tangent line to the plate, normal to the edge, and directed towards the interior of the plate, we have (-22) /cos(/î/, a)-+-^cos(/i/, p)=- -^- If the currents are permanent, we must, according to equality (21), have, at any point of the plate, and, according to equality (22), have, at any point of the edge, (24) fcos(ni, a)-i- ffcosi/ii, ^) = o. 11 It is easy to see that this last equality can still be written, designating by dl an element of the edge of the plate, (^5) ^f^-^^^dl='' Let us assume not only that the currents are permanent, but also that the plate is homogeneous-, let ^H. be the specific resistance of the material that forms it; let 8 be its thickness at the point (a, ^) Let us assume (26) tl-î 432 BOOK V. - METALLIC CONDUCTORS. Ohm's law will give us, as we can easily see, ^ -^ îl A da ' (^7) ■ - r dV Assume that the plate has the same thickness at all points; % will be independent of (a, ^). By virtue of equations (2-), equation (23) will become d /B dV\ d /A dV while equality (25) will become ^'^-^^ A You II ^" B â^ di ~ "■ If we add to these equations the condition of taking given values along the contour of the electrodes, we will have obtained all the conditions that determine the potential function V. Once this function has been determined, the equalities (27) will give the components of the surface flvix at each point of the blade. The determination of the function V becomes much simpler if we can find a curvilinear coordinate system on the surface of the plate for which A = B is constant. Such a system is called an isothermal system. It is also called an isometric system, which is due to M. O. Bonnet and which recalls the following fact: If Von agrees to always take d(^,jk), whose rectangular coordinates :r, r have for respective values 37 = a, j' = p. To any infinitely small figure drawn on the surface this law causes a similar infinitely small figure drawn on the plane to correspond. It is said that such a correspondence provides a conformal representation or a geographic plot of the surface on the plane. The angle of two lines drawn on the surface is equal to the angle of the two lines that represent them on the geographic plot. Conversely, if we have the geographical layout of a surface on a plane, any rectilinear and rectangular coordinate system on this plane is the representation of an isothermal system of the surface. These proposals show that the search for an iso (') Gauss, Allgenieiiie Auflosung der Aufgabe die Tlieile eiiier gegebeneii Flàche auf einer andern gegebenen Flàche so abziibilden dass die Abbildung dem Abgebildeten in den kleinsten Theilen àhnlich wird (Gauss, Werke, Bd. IV, p. 193). - G. Darboux, Leçons sur la théorie des surfaces, t. I, p. 146. D. - I. 28 434 BOOK V. - METALLIC CONDUCTORS. therm on a surface is equivalent to the search for a conformal representation of this surface on a plane. If, therefore, on the surface of the slide in which we want to study the permanent movement of electricity, we know how to find an isothermal system, we know, by this very fact, how to make the geographical trace of the slide on a plane. This tracing will draw a plate, whose edge and electrodes will be the images of the edge and electrodes of the studied slide. We can always suppose that the correspondence between the blade and its representation has been obtained by equating respectively k X and y the parameters a, ^ of the isothermal system taken, on the blade, for coordinate system. The function V(a, [3) will then be transformed into a function Ni^x^y')^ which will have the following properties: 1° By virtue of equality (3o), at any point on the plate, image of the blade, we have 2° At any point of the contour of the electrodes of the plate, images of the electrodes of the blade, one will have, in virtue of the equalities (Sa), 3'^ At any point on the edge of the plate, image of the edge of the blade, we will have, by virtue of equality (3i), ^-="' N,- being the normal to the edge of the plate towards the interior of this plate. Thus, the function V(a,[^), which solves the problem of the permanent motion of electricity on the blade, is identical to the function V(^, y) which solves the same problem for the plate, a conformal representation of the blade, and we can state the following proposition: When we know how to make on a plane the geographical trace of a curved blade and to find the permanent movement of V electricity in a plate cast on this trace, we know how to find CHAI". V. - MOVEMENT OF' ELECTRICITY IN A PLATE. 435 see the permanent movement of electricity in the curved blade. These theorems, whose inventor it would be difficult to name, since they are so closely related to a host of questions of analysis, were given explicitly by G. Rirchhoff(' ). Let us prove again, to finish this study, this beautiful theorem: When V electricity moves in a blade of permanent motion, the lines of equal potential level and the lines of flux always form an isothermal system. These two families of lines, equipolential lines and flux lines, always form an orthogonal system. We can therefore assume that we have taken these lines as a system of curvilinear coordinates on the surface of the blade, the a lines coinciding with the equipotential lines, and the ^ lines with the flux lines. We must always have equality ( ^\ "> /B ^\ _^ /A dW But, for the equipotential lines to coincide with the lines P = const., it is necessary that V depends only on a, which reduces equation (28) to d /B dY\ ôx \A. doL J Thus, denoting by W(^) a certain function of [i, A d% ^^^ (*) G. KmcHHOFF, Ueber die stationàren elektrischen Strômungen in einer gekriimmten leitenden Flàche {Monatsber. der Akademie der Wissenschaften zu Berlin, 19 July 1875. - Kirchhoff's Abhandlungen, p. 56). - On the relation of these theorems to the theory of analytic functions, see the already quoted work of M- F. Klein, Ueber Rieniann's Théorie der algebraischen Functionen und ihrer Intégrale (Leipzig, 1882). 436 BOOK V. - METALLIC CONDUCTORS. dV Moreover, -^ depends only on a; if we denote this function of a by $(a), we will have and the equality (19) will become "?52= X2(a,[B)[2(a)^a2H- W2(P)^j32]. Now let's make the following change of variables, a change that will not change the coordinate lines: a'= / ^(a) da, The function ).(a, [i) will turn into a function 4^(a', |i'), and we will have The line system a'=const, [B'=const. thus forms an isothermal system; and, as this system coincides with the system a = cons!., p = const., the stated theorem is proved. M. Boltzmann ("), G. Kirchhofï (2), É. Mathieu (3) have completely solved, by means of these principles, the problem of the motion of electricity in certain surfaces. We refer the reader to their works, and in particular to the Treatise of É. Mathieu. (') Boltzmann, Ueber die Bewegung der Elektricitàt in krummen Flâchen {Sitzungsber. der kaiserlichen Akademie der Wissenschaften zu Wien, t. LU, p. 2i4; i855). (' ) G. KiRCHHOFF, loC.cit. (' ) É. Mathieu, Théorie de l'Électrodynamique. Paris, 1888. CHAP. VI. - JOULE'S LAW. 437 CHAPTER YI. JOULE'S LAW. When a linear conductor is crossed by a permanent current, it heats up; the laws of this heating have been determined experimentally, starting in 1840, by Joule (' ), Lenz (2) and M. Edmond Becquerel (^). The result they arrived at can be summarized in the following law, which is called Joule's law: When a linear and homogeneous conductor, having at any point the same temperature, is traversed by a permanent current, each element ds of this wire becomes a heat source. In time dt^ this element gives off a quantity of heat whose value dQ is given by P equality (i) Ed()=-Ri'^dsdt, E being the mechanical equivalent of heat, Kds the resistance of r element ds, J the current intensity. This law can also be put in another form. Let V be the value of the potential function at the origin of the element ds ^\ (^ + ir ^^) ^^ value at the end of the same element; Ohm's law gives us RJ o?i- = - £ - - ds. as (') Joule, On the heat evolved by nietallic conductors of electricity, and in the cells of a battery during electrolysis ( Proceedings of the Royal Society, 17 December i84o; Philosophical Magazine, 3' series, t. XIX, p. 260; 1841, and t. XX, p. 204; 1843). (') Lenz, Ueber die Gezetze der Wârme-Entwickelung durch den galvanischen Strom (Poggendorff's Annalen, t. LXI, p. 44; i844) (') Edmond Becquerel, Des lois du dégagement de la chaleur pendant le passage des courants électriques à travers les corps solides et liquides {Annales de Chimie et de Physique, 3' série, t. IX, p. 21; i843) 438 BOOK V. - METALLIC CONDUCTORS. Equality (i) can then be replaced by the following: dV (■2) ^dÇl = - z-T-idsdt. G. Kirclihofr(<) has shown how Joule's law can be extended to permanent currents flowing in an extended conductor in all dimensions. Let us consider, inside a homogeneous conductor through which permanent currents flow, an infinitely loose channel whose walls are generated by a series of flux lines {fi g- 87). Fig. 87. It is natural to assimilate such a channel to a linear conductor through which a permanent current flows. Let us cut this channel by two normal sections co, to', whose infinitely small distance, MM', is equal to ds. If we assimilate our small channel to a linear conductor, the intensity of the current which crosses it at M will have the value J = iiù, i being the electric flux at point M; the resistance of the element MM will be R f/5 = - ds, Si. being the specific resistance of the substance which forms the conductor. We are therefore led, by applying formula (i) to the segment between the two sections o), w', to suppose that this segment gives off, in the time dt, a quantity of heat given by the equality E(iQ =Sltùi^dsdt. (') G. KiRCiiHOFF, Ueber die Anwendbarkeit der Formeln fur die Intensitâten der galvanischen Strôme in einem System Unearer Leiter auf Système, die zum Theil aus nicht linearen Leitern bestehen {Poggendorff's Annalen, Bd. LXXV, p. 189; 1888. - Kirchhoff's Abhandlungen, p. 33). CHAP. VI. - L.\ JOULE'S LAW. 439 But, on the other hand, if we denote by u, v^ iv the components of the electric flux at point M, we have J^a quantity iii ds is nothing else than the volume of the small segment between the sections to and w'. We thus arrive at the following statement, which is the extension of Joule's law to a conductor of finite extent in all dimensions: When a homogeneous conductor, whose points are all at the same temperature, is traversed by a permanent current, each element of volume dx dy dz of this conductor releases, in the time dt^ a quantity of heat o?Q given by the formula (3) Ed<^ = S^{u'^-^v'^^w^)dxdydzdt. This equality can be transformed as we have transformed equality (i); indeed, Ohm's law gives ox S\,v = - £--, at ^ (V = - s -r- . dz According to these relations, equality (3) can be written : /c)V d\ ôW \ (4) ^d^l^-zi^^u-^ -v-^^w'jdxdydzdt. R. Clausius (') gave a statement of equations (2) and (4) which is interesting to know. The electrostatic potential of a system having the expression expression in which the summation extends to all loads (-) R. Clausius, Ueber die bei einem stationâren elektrischen Strome in dem Leiter gethane Arbeit und erzeiigte Wàrme {Poggendorff's Annalen, Bd. LXXXVII, p. 4i5; i852. - Mémoires sur la Théorie mécanique de la chaleur, translated by Folie, t. II, p. ii4). 44o BOOK V. - METALLIC CONDUCTORS. of the system, when the electrical distribution undergoes an infinitely small variation, this potential experiences a variation (5) o\y = z\ègèg, ùq being the variation of the electric charge at the point where the potential function has the value V. This being said, let us first consider a segment MM' of linear and homogeneous conductor, crossed from M to M' by a permanent current of intensity J. In time dl, this segment releases, according to equality (2), a quantity of heat c/Q given by ErfQ = - z(y-\)idf. Let us imagine that, during time dt, the segment MM' had been traversed by the uniform current of intensity J, but that the electricity had remained at rest on the rest of the system. The electrical distribution on the system would then have undergone a change in time dt; the charge at point M would have decreased by J dt, and the charge at point M' would have increased by the same amount. According to equality (5), the electrostatic potential of the system would have increased by 8W = e(V'- V)J^;. The comparison of the two equalities we have just written leads to the following proposition: Let MM' be a linear, homogeneous segment of a conductor through which a permanent current flows. It gives off, in time dt, the quantity of heat dQ which can be calculated in the following way: let us suppose that, in time dt, the segment MM' has been traversed by the permanent current which in reality traverses it, while the rest of the system has not been traversed by any current. The electric distribution on the system would have undergone a certain change, resulting in a certain variation oW of the electrostatic potential of the system, and Von would have (G) E^Q=- ûW. This theorem can be extended to three-dimensional conductors as follows: CHAP. VI. - I.A JOULE'S LAW. 44' Inside a homogeneous conductor, having at any point the same temperature, through which permanent currents flow, let us trace a closed surface S (^fig- 88). The part of the conductor if The heat generated inside the conductor releases, in the time dt^ a quantity of heat û^Q, and we have, according to the equality (4), ErfQ =-srf/yyyy(-^"H- ^.-^ ^\.) dxdydz, the triple integral extending to the volume bounded by the surface S. Let JN/ be the normal to the surface S towards the interior of this surface. An integration by parts gives : = - V V[?f cos(N/, x)-^ V cos(N/,_7) -+- w cos(N;, z)dS If we notice that, the currents being permanent, we have, at any point inside the surface S, du ôv ôw T" -^ T" + X" = ^' ox oy az we see that we can write (7) Efi?Q = £C^^ C V["cos(N,-, ar)4-pcos(N/, 7)-r- tv cos(N/, z)] rfS. Let us imagine, on the other hand, that the space inside the surface S remains crossed, during the time dt^ by^ the permanent currents which cross it in reality, while the electricity would be in rest on the rest of the system. The electrical distribution on the 44" BOOK V. - METALLIC CONDUCTORS. system would undergo a certain change that we can determine. The electric charge at a point outside the surface S would not change, since there would be no current at that point. The electric charge at a point inside the surface S would not change either, since at this point the current would be uniform. But it would not be the same at a point of the surface S. The element c/S of the surface S would acquire, in the time dt^ a charge 0^ = - dt\u cos(N/, a-) -+- f cos(N/, y) -t- w cos(N,-, z)\ dS. This change in electrical distribution would lead to a change in electrostatic potential, having as value, according to the formula (5), (8) SW ^- - zdt C V[i the quantity - is in general finite, but that the quantity - ^ is always extremely small. Finally, let us observe that Vo[cos(r, n/)] = i, Vi[cos(r, n/)] = cos (r,n/), 454 BOOK V. - METALLIC CONDUCTORS. and we will have Taking into account equality (i), if we observe that cos(r, rii) _ /■ /'■^ Ofii if we finally put (2) OIl^S -- C p / rf^-, we will have V = D1L - ^S. (Jrii If it is a question of calculating the potential function, not of the small element A, but of the whole layer, we will have for expression of this potential function ô^ (3) Y=^DrL~dS. This expression is valid only as long as the point M, to which the function V refers, is at a large distance from the surface S with respect to (). H- jx). DÏL will be named [ intensity of the double layer at a point of the element dS. Let us assume that the double layer considered is the one that is found, in the state of equilibrium, at the separation surface of a conducting body A and an insulator o. In equality (2), we must make dif = dl dS, P = p(A,o, /), and 0 will be defined by the equality 01I(A,o) = / /?(A,o, l)dl, or (4) D\iiX,o)---= -- / lAe(k,o,l)dl. 4Tr£ J^ CHAP. VII. - POTENTIAL DIFFERENCE IN CONTACT. .05 This equality shows us that the quantity DTL(A, o) is completely defined when we know the nature of the metal k. and of V insulator o. Let us consider in the same way the double layer which forms, in the state of electrical equilibrium, at the separating surface of two metals A and B. Let us suppose that the direction, designated in the foregoing by /i, is the normal "^ towards the interior of the metal. We will have again dv = dS dl; l will have to vary from - ()v -+- (jl) to H- (X -h a ); / varying from - (^ + [J"-) to o, p will take the determination p(B, A, - ■ l); on the contrary, / varying from o to ()v -f- [x), p will take the determination p(A, B, /). We will thus have OTL=:/ lp{B,X, - l)dl-i- lp{X,B,t)dl, or i lp{k,B,l)dl- lp(B,A,l)dl, what we can still write (7) 3rt(A,B)--l: 4710 J^X,+ [JL ^X+[i "I f l^e{X,B,l)dl- lù.e{B,A,l)dl\ ^^0 .1 This equality shows us again that the quantity 31L(A, B) is fully defined when the nature of the two metals A and B is known. These two quantities, 311 (A, o), 31L(A, B), are still susceptible of another expression. Consider first the quantity D1I/(iV, o). The quantity varies very quickly, in the direction of the normal to the surface S; at on the contrary, its variations are very slow following a parallel to the tangent plane to this surface. We can therefore reduce significantly ACk/ A A ' <^2e(A,o, /) Ae(A, o, /)a ^^^ We will thus have ^'^■°>=4^eX "/. *- 456 BOOK V. - METALLIC CONDUCTORS. An integration by parts transforms this equality into 4TrE L (^t Jo 4to Jq àl If we observe that and if we integrate the last term, we find (8) OrL(A,o)= -^[e(A,o,o)-0(A,o,X-h[x)]. In a similar way, equality (7) can be transformed into 3K(A.B)= ■ M''(A.B.^, _,ç "e(g^y. 4^£ L"^ "' Jo If we observe that rc)e(A ,B, /) 1 ^^ peçB, A, /) ] _ 0(A, B, o) = e(B, A,o), the previous equality will become (9) 01I( A, B) = -1- [e(B, A, X + fx) - 0(A, B, X -^ ^)\. 4 ir£ Let us return to the expression of the potential function of any double layer, given by equality (5). Suppose that the surface S is closed and that the double layer has the same constitution at all points of this surface. The quantity Oit will then have the same value at all points of the surface S, and the equality (5) can be written But Gauss's lemmas teach us that at any point M, CHAP. VU. - POTENTIAL DIFFERENCE AT CONTACT. 4'^7 outside the surface S, we have at and that at any point M, inside the surface S, we have We have therefore at any point M, outside the surface S, and whose distance to this surface is large compared to X, (10) V = o and at any point M, inside the surface S, and whose distance to this surface is large compared to ), (11) V=:47rDR. Let us make some applications of these equalities (lo) and (i i). i" Consider a homogeneous conductor. The natural distribution forms on this conductor a homogeneous double layer {see p. 3-6) for which OÏL is given by one of the equalities (6) or (8). The potential function of this natural distribution will have the value o outside the conductor and the value 4TtDlL(A, o)= i[e(A, o, G) - e(A,o, X^ (x)]. This is what we had already found previously [Chap. II, equalities (5) and (6)]. 2° Let us consider a heterogeneous conductor, formed by two metals A and B. If we denote by 8^^,07 Sgo the two portions of closed surface which limit the conductor, and by S^g the contact surface of the two metals, the equality (5) can be written more explicitly d'- d^ d' V = D1I(A, o) j^ ^ dSA,o^ 3n,(B, o) g ^c?SB,o-i- 31I(A,B) g ^ ^Sa.b This expression can be simplified, by means of the equalities (8) and (9). V 4Tr£ 458 BOOK V. - METALLIC CONDUCTORS. If one obsei've that one has e(A,o, ^) = 0(A, B,). -t- IX) = e(A), e(B, o, X -f- [ji) = e(B, kA-v n.) = e(B), we can write -l-[e(A,o,o)§-i S r!i< '^"1 ■ Let's put e(A,o, o) - e(B,o, o) I-- -, and ^1^ ^ e(A,o, o)- p. ^ [X - e(B, o, o) ^ 47:2 i^-Kt If we observe that, for any point outside the conductor, we have ) =0, CllAP. VII. - POTENTIAL DIFFERENCE AT CONTACT. 4^9 we can write = 3'^S3) VCMa)4-£0(A) = /(TTô u dl Similarly, for a point Mj, inside the metal B, it gives (i3 bis) V(MB) + £e(B) iT.Z d' 0(A, o, o) V - c?Sa,o e(B,o,o)S-^£^SB," ] The second members of these equalities are obviously not constant. To be sure of this, it is sufficient to note that V - - dSj^^^ and V-- t/Seo are the angles under which, from the point M, we see the inner faces of the surfaces S^^o, Suo- Now, for the natural distribution on the two metals A and B in contact to be an equilibrium distribution, we would have to have V(MA)+£e(A) =V(MBJ-+-£e(B) = G, G being a constant. The two metals in contact must therefore be electrified. If we observe that S5^/^'^'""^S^-^'^'"=^°' CHVP. VII. - DIFFERENCE IN POTENTIAL ON CONTACT. î6l it is easy to see that V eleclriché ■ conifnunrq née will have to be distributed in such a way that we have, inside the conductor, e(A,o, o) - e(B,o,o) à' Sâ;::-^ c. 4 710 The study of this distribution, made by any means, can never make known that the quantity e(A,o,o) - e(B, G, o), not the quantity e(A) - e(B). § 3 Permanent currents in metallic conductors heterogeneous. The preceding considerations apply only to the case where the electrical equilibrium is established on heterogeneous metallic conductors. We will now study the case where these conductors are crossed by permanent currents. In order to obtain the fundamental laws of permanent motion of electricity in heterogeneous metallic conductors, we will extend to them the first fundamental assumption concerning the steady state, an assumption which, in the case of homogeneous conductors, is only another form of Ohm's law [Chap. IV, equality (2)]. Let us consider the set of variables, other than the electric fluxes, which define the state of the system at time t (what we call the system supposed to be currentless). To this system supposed to be currentless we can apply the consequences of Carnot's principle; any virtual transformation of this system corresponds to a certain amount of uncompensated work. Let {x,y, s) be a point in the system and(;z: -f- ox,y-^ Sy, z -{- ùz) an adjacent point. If an electric charge dq passed from the first point to the second, some uncompensated work would be generated in the system. Let us denote it by d'z. Let us denote by C^, ^y, Czthe components of the electromotive force at the point (cc,y,z). By hypothesis, we have ( l4) d-z = (C.c ox -i- Cy8y -i- Cz 0^) dq, whatever Sx, 0JK5 ^^ 46-2 BOOK V. - METALLIC CONDUCTORS. Now we can calculate d-z. The system assumed without current has an internal thermodynamic potential §. The change of position of the electric charge dq does not move any part of the system and, consequently, does not cause any external work. We have therefore (i5) d-^ - li. Moreover, we have [Liv. IV, Ghap, II, equality (i5)] the various letters that appear in this formula having a meaning that has been explained in the place quoted. Since the displacement of the charge dq does not cause any change in the physical or chemical state of the various conductors of the system, we see that the variation undergone by the internal thermodynamic potential is reduced to (i6) / \ -^ l ' \dx dx "^ ~^ dx " ) ^' The equalities (i4), (i5) and (i6) having to take place, whatever ox, hy, 8^, give dx âx ' ('7) ^y--^ -^)j, ^^ Cz = - £ dv of dx âx dY from dy d\ __ from dz toz These equalities give the components of the electromotive force at a point of a heterogeneous metallic conductor whose constitution varies in a continuous way and which is crossed by T dB de de permanent currents. The terms r- ? -j-, - -r- represent r dz dy dx ' the correction that must be made to Ohm's law in the case where the driver is not homogeneous. If we denote by Si. the specific resistance of the conductor at the point with coordinates x, y, z and by ", p, w the components CHAP. VU. - POTENTIAL DIFFERENCE AT CONTACT. 463 of the electric flux at this point, the equalities (17) will become (.8) Let's see how electricity is distributed inside a similar conductor. Let's divide the two members of each of the equations (18) by cîR; let's differentiate the two members of the first one with respect to x, the two members of the second one with respect to y^ the two members of the third one with respect to z^ and let's add member by member the results obtained, observing that [^-=-^Tx dx 1 dV from from dz' du dv div dx dy dz We find i^(sAV^Ae) '9) to- d- to- o. T / c)A dSi. dcfl M -T \-V - h tV -- 4t:£ \ dx dy . dz \ dx dx dy dy dz dz If denotes the electric density at the point (^, J', -s), we have AV = - 4itp, and equalities (18) and (19) give (20) o = -^Ae - 4lTS while, in the equilibrium state, the electric density at the same point would have the value p = -^ Ae. In the interior of a heterogeneous metallic conductor carrying permanent currents, the density does not have the same value as in the steady state. It is only in the interior of the conductor that the 464 rjvKE V. - metallic conductors. The two values of the homogeneous conductor become equal to each other, and equal to o. Let us now consider a discontinuity surface separating two regions a and b where the conductor has different properties ("■- 92)- _ ■ _ _ We know that at any point M of this surface we must have (-M) Ua COS(N", X) -+- Va. COS( Na, y) -+■ Wa COs(Na, z) uo cos(N6, op) -+- v/, cos(N6, y) -\- w/, cos(Nft, z) - o, or, by designating 'by ia the component following N^ of the flux at Fig. 93. inside the conductor a at a point infinitely close to the point M; by i^ the component along N^ of the flux inside the conductor b at a point infinitely close to the point M, {-21 bis) ia-T- ib= ()■ The normal component of the flux has the same value on both sides of the surface. Is the same true for tangential components? Through the point M, let us lead two directions T, T', rectangular between them and located in the tangent plane to the surface S. At a point of the conductor rt close to the point M, the flux /"" has as components, along MT, MT', ta and t'^; at a point of the conductor b close to the point M, the flux fh has as components, along the niAP. seen. - POTENTIAL DIFFERENCE AT CONTACT. 465 same lines, ti, and ^^. Equalities (i8) give The well known properties of the first derivatives of the potential function give dT dT' dT dT' On the other hand, by hypothesis, the first order partial derivatives of the function B vary in a continuous way, even at the passage of a discontinuity surface, which gives So we have (22) The electric jiux on either side of a discontinuity surface are in the same plane as the normal to the discontinuity surface. The projections of these two flows on the tangent plane to the surface have the same direction. From the comparison of equalities (21a) and (22) we deduce the equality S^a tang(/a, Na ) -+- iKi, tang(//" N^) = o, which indicates how the direction of the electric flux varies when passing a discontinuity surface. The set of propositions that we have just demonstrated has received from Maxwell (' ) the name of the laws of the refaction of the electric flux at the passage of a surface of discontinuity. (') Maxwell, Traité d'Éleclricité et de Magnétisme, translated by G. Seligmann-Lui, t. 1, p. l\<.yi. Paris, i885. D. - I. 3o d^a dt dT' dSa dSb dT dT l J^a ta - ■ Slt>t/j= 0, { ^at'a~ ■ài'/J',,--- 0. 466 BOOK V. - METALLIC CONDUCTORS. Returning to equation (21), qvii can be written, by virtue of the equalities (18), OR J_ / àV_ àY_\ f d(da , àei,\ I \ \ d If we denote by t the surface density at the point M of the discontinuity surface, we have - 4Tra, inN'a to&a ^ àeo to^a àNh _ - (sV + 0) - - <îR/,4 - <^64, and the previous equality will become Thus, at the separating surface of two metallic conductors a and h^ through which permanent currents flow, there is a surface electrical distribution whose intensity at each point is proportional to the excess of the resistance of conductor a over the resistance of conductor b and to the component, along the normal to the suif ace, of the flux that penetrates conductor a. Similar considerations apply to the separation surface of a conductor a and the insulator i. In this case, the electric flux is zero at any point of the insulator i, and moreover we have, at any point of the separation surface, tfaCOS(Na, X) H- Ç>aCOS(Na,7) -f- "'aCOS(Na, s) = O, which gives, according to the equalities (18), The electrical density at a point on the surface separating a conductor from an insulator is therefore, if we denote by N^ the normal CflAP. VII. - POTENTIAL DIFFERENCE AT CONTACT. 4^7 external to the driver, L ^X _^ _i_ -^ When the constitution of the conductor is known, equation (19) becomes a partial differential equation which, together with the boundary conditions (aS) and (sS), determines the function V. Once this function has been determined, the equalities (i8) determine the components of the electric flux at each point. Let us apply the preceding considerations to a conductor through which permanent currents flow, in which two metals a and b, homogeneous up to a small distance from the end surfaces, are in contact with each other. 1° At a distance greater than (7^+ [x) from the surfaces that bound each of the two metals, there is no free electricity spread inside these metals; the equations of motion of electricity are those given by Ohm's law. 2° In the vicinity of the surface that separates the two metals a and b^ there is electricity spread inside each of the two metavixes. The distribution of this electricity depends on the intensity of the current flowing through the conductor. 3" On the surface itself, there is a surface distribution whose density has the value (27) G= ■-^[Aa(0)-Sih{0)]ia, Sla{i), ^^b{i) being the specific resistances of metals a and 6, at a distance / from their contact surface. The electricity thus distributed on the contact surface of two metals through which permanent currents flow no longer forms a double layer, as in the steady state. To prove this theorem, let us agree to represent by 0(a, Z/, /), <îfl(rt, ^, /), f(rt, 6, /) the values of ©"j^l^, ia^ at a point of the conductor a located at the distance / from the surface that separates the conductor from the conductor b. At this point, the electric density 4()8 BOOK V. - METALLIC CONDUCTORS. has the value, according to the equalities (20) and (18), ('^8) p(",è, l) ^ - !- Ae(a,6./) ^i_ I J dSl(a,b,l) d[ z\-^e(a,b,l)] 4TrE S{.{a, b, L) \ dx dx dSl(a,b,l) d\zY ^%{a,b,l)\ dy ^ dSi{a,bJ) d{sY -^e (a,b, l)] dz Let us lead two surfaces parallel to the surface S {/ig'- g'i), which separates the two metals a and b, these two surfaces being located at Fig- 93AL _Tr" a distance slightly greater than (). - [*■) from the surface S, one, S^, inside. the conductor "; the other, S^, inside the conductor b. On the surface S, let us take an element MN, of area dS. Through all the points of the contour of this element, let us lead normals to the surface S. These normals detach on the surface S^ an element M^N.i and on the surface Sô an element M/, Ni . Let us find the quantity of electricity contained in the closed surface M^N^MiNj, neglecting in this calculation the ratio of Çk -\- u.) to the radii of curvature of the surfaces S", S^. The element dS carries a quantity of electricity which has the value, according to equality (27), 0,^=-' - [A{a,'b, o)~éfl(b, a,o)]i(a, b, o). The MNAI^Na surface contains, inside, a quantity of electricity i p(a,b,l)dl. CH.VP. VII. - CONTACT POTENTIAL DIFFERENCE. 4^9 The MNMôNô surface contains inside a quantity of electricity Qb=^dS ç>{b,a,l)dL The quantity we want to calculate has the value If we refer to equality (28), we can easily see that we can write p(a, b, l)^^ ~ Ue(", b, l) -f- i{a. b, l) '^i'Î^^lM]] and also So we have Q==-f^ / [Ae(". 6, /)-4-Ae(6,a, 0]^^ ,"/ /, Nv / ^ r^'''^dSl{a,b,l) ., , ,, -^ S{.{a, o, o)i{a, b, o) -^- 1 -jj - - - i{a, b, i) al -~S\.{b,a, o)i{b, a,o)-+- j t(6, a, l) dl\ . The quantity represents the amount of electricity that would be present on the MaNaM^Ni surface, if no current flowed through the system. In this case, the electricity distributed near the contact surface of the two metals would form a double layer. The previous quantity is therefore equal to o. An integration by parts gives us f i (a, b, l) al âl 9 = -/" ' Sl{a,b,l)' dl dl -r- i{a, è, X -f- \x)S{.{a, 6, X h- [j.) - i(a, b, o) Sl(a, b, o). But it is easy to see that the integral in the second member is of the order of (X-h[x) and can, therefore, be 470 BOOK V. - METALLIC CONDUCTORS. neglected. We have therefore J"/H-U. f t(") b, l) dl and, similarly, i{b,a,o) ël{o,a,o) -T- I i{b,a,l) ^^ - = i{b. "(X -f- [x) S\.{b, a, X -+- p.). If we notice that i(a, 6, X -i- jx) = i{a, b, o), i( b, a,A -h [i) = i(b, a, o), to quantities of the order of (X + [x), and that, moreover, according to equality (21 bis), i{a, b, o) -r- i{b, a, o) = o, we see that we will have dS ('9) ( Q = ^~ [Siia, b, l -^- [i) - A {b, a, l-r- [i)]i{a, b,o) < = - - [^ib, a, X -r- [x) - cR(a, 6, X -f- [x)]t(6, ", o). If the electric flux is not tangent to the discontinuity surface, and if the specific resistance is not the same inside the two conductors, this quantity cannot be equal to o. It is interesting to compare the results we have just obtained, by studying a heterogeneous metallic conductor carrying permanent currents, with the results we obtained in § 1, by studying the electrical equilibrium on a similar conductor. When the electrical equilibrium is established on the system that encloses the two metals a and è, the electrical density at a point on the contact surface has the value ff = 0, and the amount of electricity contained in the volume MaN^MôNô a for value Q = o. When the system is traversed by uniform currents, one CHAP. VII. - POTENTIAL DIFFERENCE AT THE CONTACT. 47 1 a, by virtue of equalities (17) and (19), !T = - - [.^(a, 6, o) - ifl(6, a, o)]f(", 6, o), Q = 7 - [<îR.(a, 6, X -t- a) - Jl(6, a, X -i- ijL)]t(", 6, o). 47î£ The analogy of the expressions obtained for a and Q in the first case is found in the second. Suppose that ABCD i^fig. 94) is a homogeneous metal a than at the distance (X -i- |x) of the surfaces which terminate it. By the face AB, this metal borders on another metallic conductor, formed by a metal b^ which maintains all the points of the surface AB at the same potential level V,. Likewise, by the face CD, it borders on another metallic conductor, formed by a metal c, which maintains all the points of the surface CD at the same potential level Vo. The surface ABCD confines with the insulator. Let us propose to find the distribution of the uniform currents that flow through this conductor. This is a problem that we have already dealt with, assuming that the conductor is homogeneous up to the terminal surfaces. Let us lead a surface A'B'C'D', located inside the conductor a, at a distance i^-^[a) from the surface ABCD. Let M be a point on AB. Through this point, let us lead a normal N^ to the surface AB towards the interior of the body a. It meets at M' the surface A'B'. Let V'^ be the potential level at M', a level that we propose to determine. At a point between M and M', at a distance / from the point M, we have, keeping the notations we just used Sl(a,b, l)i{a,b,l) = - t-jj- - ■ -^ 472 BOOK V. - MKTAI.LtQUES DRIVERS. and, therefore, £(Vi - V;)-4-e(a,6, o) - e(",6, X-^[x) = -- / ^(",6, l)i{a, b, l)clL The second member is a quantity of the order of Q^-i-'^)', it is, therefore, negligible; therefore all points of the surface A'B' are at the same potential level V'i = Vi -+- ^ [&{a, b, o) - e(a, b,\-^ [Ji)]. Similarly, all points on the surface CD' are at the same potential level V'j =^ Vs-h - [6(a, c, o) - 0(a, c, X --t-[x)]. Let P be a point on the surface AC, through which the conductor a borders the insulator. Through this point, let us lead a normal PN' to the surface AC. It meets at P' the surface A'C At point P, we have u cos(N', x) -i- V cos(N', y) -^ w cos(N', z) - o. Because of the continuity of the components of the electric flux, at the point P', the quantities u' , v\ w' will differ from ", c, w only by quantities of the order of (\-'\- ]x). One will have thus appreciably u! cos(N', x) -f- v' cos(N', y) -- w' cos(N', z) - o. But the point P' is in the homogeneous region of the conductor where Ohm's law becomes applicable, so that the previous relation can be written M' = ""■ This equality will be verified at any point on the surface A'C'B'D'. Finally, inside the closed surface A'C'B'D', the conductor being homogeneous, we have AV =-: o. The function V is therefore harmonic at any point inside the closed surface A'C'D'B'; it takes, on the surfaces A'B', CD', given values Vj, V'^ -, on the surface A'C'B'D', it verifies the equality m' = ^ cn\P. VII. - POTENTIAL DIFFERENCE AT CONTACT. 47^ It can be seen, therefore, that the determination of the currents flowing in our conductor at a distance greater than {\ -h jj.) from the terminal surfaces will be obtained precisely by the analytical method formed in Chapter IV, for determining the currents flowing in a homogeneous conductor up to the terminal surfaces. Moreover, once the components of the electric fluid are known up to a distance (1+ pi) from the terminal surfaces, it will be easy, by continuity, to know their values jto the terminal surfaces. The result we have just obtained shows that the consequences to which, in Chapters IV and V, we were led by applying Ohm's law to absolutely homogeneous conductors, remain valid for conductors which lose their homogeneity at a very small distance from the terminal surfaces. § 4 Method of M. Pellat to determine the potential level differences of two metals in contact. The study of electrical equilibrium and of the permanent motion of electricity on heterogeneous metallic conductors would still involve the examination of a great number of questions which it is not possible to deal with here. We shall therefore confine ourselves to the principles set forth in the preceding paragraphs, and for an examination of the general consequences of these principles, we shall refer to a Memorandum that we have published on this subject (' ). Nevertheless, before abandoning this order of questions, we will expose the method followed by M. Pellat (2) to determine the potential level dilTerence of two metallic conductors in contact. Let's take the two concentric spherical conductors that we have already considered in § 1, and that, in practice, we (') P. DuuEM, Sur la pression électrique et les phénomènes électrocapillaires; première Partie, De la pression électrique {Annales de l'École Normale supérieure, 3" série, t. V, p. 97; 1888); seconde Parlie, Des phénomènes électrocapillaires {/bid., 3° série, t. VI, p. iS3; i88()). (') H. Pellat, Difference of potential of the electric layers which cover two metals in contact {Annales de Chimie et de Physique, 5" série, t. XXIV, p. 5; 1881). 474 BOOK V. - METALLIC CONDUCTORS. replace them with two parallel plates. Let us imagine that a wire MN with a bend in it is added to these spheres {fig. po). From the point m of this wire, which is made of any metal, another wire /??/>, also made of any metal, leaves and rejoins the The conductive sphere 2, formed by the substance a, is connected at p. Similarly, the conductive sphere 4? formed by the substance h is connected by a wire of any kind p' n to the point 11 of the wire MN. Let's assume that the steady state is established on the system. First, I say that in this steady state, there is no current outside the MN wire. If there were a current in the wire mp whose intensity differs from o, it would enter, during the time dt in the part of the system formed by the wire mp and the conductor 2, a quantity of electricity jdt which would not leave it. It would be necessary, therefore, that the electrical distribution be modified on the considered part of the system. Thus, the wire mp is not crossed by any current, and it is the same for the wire np' . Can it happen that permanent currents exist either in the ground of conductor 2 or in the ground of conductor 4? As we have just seen, these conductors can be considered as isolated. Let M, ç", w be the components of the electric flux at a point of an isolated system formed by homogeneous or heterogeneous metallic conductors. CHAP. VII. - POTENTIAL DIFFERENCE AT CONTACT. 47^ INoiis will have, according to the equalities ('j), which can be written [^ ox oy oz d(£V-4-e) , ()(£V-T-e) , _àizY-^6y Let's integrate the two members of this equation for the space occupied by the set of conductors 2 and 4" and we will have I j jSi{u^--^v^'i-w^-)dxdydz rrr\ d{z\-^e) (^(ev-^e) ()(£V4-e)i , , , An integration by parts allows to transform the second member into - C (ôV-H 0)[mcos(N/, x)-^v cos(N,,jKj -4- w cos(N/, z)] dZ. The first sign of integration extending to all the metallic parts whose nature varies from one point to another in a continuous way, and the second to all the surfaces which limit these various parts, one deduces easily from there I j I Si(u^~v^-^w^-)dxdydz ^^-ffJ{t\-^e)^^dxdydz-^{zY-^e)~d^. In the second member, the first summation extends to all electrified volumes and the second to all electrified surfaces. But, if the steady state is established, we have, at any point, dp di 476 BOOK V. - METALLIC CONDUCTORS. and the previous equality requires that we also have, at any point, u - o, (^ = o, w - o. Thus, on an isolated system, formed exclusively of homogeneous or heterogeneous metals all at the same temperature, there can be no other permanent regime than V electrical equilibrium. If we apply this general theorem to the case at hand, we see that the conductors (2) and (4) are in electrical equilibrium. Let m! be a point of the conductor MN infinitesimally close to the derivation m; let n' be a point of the same conductor closely following the derivation n; let U(m'), U(n'), 0(m'), ©("') be the values of the potential function and of the function at these points; let R be the resistance of the segment m' n' and J the intensity of the current flowing through this segment. We will have RJ = sU(m') -+- e(m') - eU(w') - <è{n'). Inside the conductor 2, we have EV2-^-0(a, i, À-f-[Ji) = £U(7n')--t-6("2'), and inside conductor 4 we will have sV^H- 0(Z>, î, À -4- ]x) = £U(rt') + e(n'): hence the relation £(V; - V'4'i-^e(a, t, \-^ p.) - 0(è, i, X -f- [Ji) - RJ. For our two spheres not to carry electricity, it is necessary and sufficient, as we saw in § 1, that the two surfaces S2 and S;) be at the same potential level T, i.e. that we have both sV'a -4- 6(a, i, X -+- fz) - sT - "-- 0(a, ï, o). eV'j -t- 0(6, i, X -H p.) = eT -f- 0(", J, o) and, therefore, according to the previous equality (20) 0(", i, o) - 0(6, J, o) -- RJ. If we therefore adjust the resistance R in such a way that, in the previous experiment, the inner sphere takes no load CHAP. VII. - POTENTIAL DIFFERENCE ON CONTACT. 4/7 (which an electroscope will always allow to see), the determination of this resistance R and the current J will allow to determine the difference (-)(", f, o) --0(6, i, <)). This procedure does not allow, any more than the one indicated in § 1, to determine the difference e(a, i, 1 -r- [x) - 0(6, i, y^ -+- [J-) The experimental precautions that must be taken when using this process can be found in M. Pellat's Memorandum. 478 BOOK V. - METAL CONDUCT. CHAPTER VIII. THE PELTIER EFFECT. § I. - The Peltier effect. We have said that the study of the permanent motion of electricity follows entirely from two fundamental hypotheses: one, which is arrived at by generalizing Ohm's law, has been indicated in Chapter IV; the other, which is obtained by generalizing Joule's law, has been indicated in Chapter VI. In the previous chapter, we applied the first of these hypotheses to a conductor made of different metals that all have the same temperature, and we deduced a series of consequences on the laws of permanent currents in a similar conductor. We will now apply the second hypothesis to this same conductor. Let us consider a closed surface S traced inside a heterogeneous metallic conductor, all the points of which are at the same temperature, and which is crossed by permanent currents. In the time dl^ the part of the conductor that is enclosed by this surface gives off a quantity of heat ûfQ that we want to calculate. For this, under the assumption indicated in Chapter VI, we can operate as follows. We will assume that, during time dt^ the currents that flow through the part of the system enclosed within the surface S are maintained, but the currents that flow through the rest of the system are reduced to o. This would result, during the time dt, in a change of electrical distribution on the system. This change consists, as is easily seen, in bringing onto the = - £ If then, in accordance with the notations indicated elsewhere [Book IV, Chapter II, equalities (16) and (19)], we pose (3) Kr^e+HT, T being the absolute temperature, we can replace the equality (2) by the equality / E o?Q -r ^("2 _4- t.2 ^- w2 ) dx dy dz dt i - T M \-v-^ h w -- ] dx dy dz dt . \ dx dy dz / -^ When a heterogeneous metallic conductor, whose points are all at the same temperature, is crossed by permanent currents, each element of this conductor gives off, in time dt, a quantity of cJialeur which is the sum of two parts: dY from dx dx dV d& dy dy dV dQ dz 'dz CHAP. Wine. The first part THE PELTIER EFFECT. 48 1 i3 .^(M^-i- t;2-|- w-)dx cly dz dt is given by Joule's law; it is always positive; it is proportional to the square of the electric flux at a point of Vêlement and to the specific resistance at this point. The second part dx Oy - 1 dx dy dz dt dz j -^ would be equal to o if the element were homogeneous; it changes sign if the direction of the electric flux is reversed at a point of the element; it is, in size, proportional to this flux; it depends on its direction. This second release of heat is called Peltier's phenomenon or à^efj'et Peltier. It is in fact Peltier (') who, in 834, was the first to demonstrate the particular calorific phenomena that heterogeneous conductors present. Let us see how one can deduce from the preceding law consequences that can be verified experimentally. Let us consider a conductor {fig. 96), formed by two metals a and ^, welded together along the surface AB by means of a metallic weld of any kind. To metal a is attached a wire F, of any nature, carrying a uniform current of intensity J. This current is carried by a wire F', of any nature, fixed to the metal b. _ ^ - - ^ ^ - ^ ^ ^ - (') Peltieu, Nouvelles expériences sur la caloricité des courants électriques {Annales de Chimie et de PJi,ysique, i" série, t. LVf, p. 871; i834). D. - î. 3i I (5) 4B'Jl BOOK V. - Li:S METAL CONDUCTORS. Noti-e conductor is flowing with uniform currents. Let us lead, inside this conductor, two surfaces normal to the flux lines: one MN, or S, inside metal <2, the other PQ, or S', inside metal b. Each of these surfaces has this property that (£V+ 0) has the same value at all its points. Each of them normally intersects the conductor surface. Let us apply equality (i) to the closed surface S or MPQN, replacing K by (0+TH). The part of the conductor inside this closed surface will release, in time dt, a quantity of heat dÇ) given by But, at any point on the surface where the conductor meets the insulator, we have u cos(N;, x) -\- V cos(N/, y) -+- w cos(N/. z) - o. The previous equality becomes EdQ= dt^(z-he + TH)[ucos{Ni,x)-{-vcos{Ni,y)-h-wcos{Ni,z)]dS + dt ^ (£V+e-f-TH)[K cos(N,-, x)^v cos(N/, y)^w cos(N,', z )] dS'. At all points on the surface S, (sV-i-©) has the same value, which we will denote by (eV+ 0); at all points on the surface S', (eV+0) has the same value, which we will denote by (£V'+ 0'). Moreover, we have J - X [" cos(Nj, X) -\- V cos(N/, y) -^ w ces (IN/, z)] dS == - ^[u cos(l\j, x) -{- V cos(N/, y)-^iv cos(Ni, z)] dS'. Thus we have 1 Q (sV-i- e)[u cos(N/, x)-i-v cos(N;, jk) -H iv cos(N/, z)] dS (6) \ -^ C (£V+0)[itcos(N,-, x) -+- PCOs(N/, y)-~ w cos(N,', z)] )TJ(/". This is the expression of the amount of heat released by a uniform current in a section of conductor formed by two welded metals. If the MNPQ section is taken large and short, its resistance R will be an extremely small quantity. If the current is not very high, we can neglect the term RJ- and simply write (he) Ed(l^{ha-hb)'Y]dt. Thus, when a current of not very great intensity crosses the weld of two metals, it gives off a quantity I 484 BOOK V. - METALLIC CONDUCTORS. of heat which depends only on the nature of the two metals at a certain distance from the end surfaces and does not depend on the shape and size of the contact surface or on the nature of the weld. This calorific phenomenon is, in size, proportional to U current intensity. It changes sign if Von reverses the current. This is the phenomenon first observed by Peltier by means of the following experiment, which can easily be repeated in the classroom. The strongest effect is obtained with bismuth and antimony. The current gives off heat when it passes from antimony to bismuth and absorbs heat when it passes from bismuth to antimony. Two antimony-bismuth solders are placed respectively in the two balls ", b of a Rumford ihermoscope {fig. 97). Fii 97 The same current flows through both welds, but in ball a it passes from antimony to bismuth and from bismuth to antimony in ball b. A heat release takes place in ball a and an absorption in ball b. The two effects add up to move the index I of the thermoscope from ball a to ball b. If the current is reversed, the index moves in the opposite direction. Let us consider three metals a, b, c; a current of intensity J, passing from metal a to metal 6, produces in time dt^ a release ciiAP. VIII. - the peltier effect. 485 of heat o?Q, given by ErfQ, = T(/ta- hb)i dt. This same current, passing from metal b to metal c, produces, in the same time, a heat release C/Q2 given by ErfQj^ T{hb-hc)i dt. Finally, this same current, passing from metal a to metal c, produces, in the same time, a heat release c/Qa given by '^dÇii^T{ha~hc)ldt. It is easy to see that we have This relation shows that, in the experimental study of the Peltier effect, it is not necessary to study the welds that can be formed with all the metals taken two by two; it is sufficient to study the welds that can be formed by joining successively each of the metals with the same standard metal. The preceding relation will then show the Peltier effect that occurs when any two metals are welded together. The experimental study of the Peltier effect has given rise to many recheixhes that one will find exposed in the treaties of Physics. § 2 - Relation between the Peltier effect and the potential level difference between two metals in contact. We will give the name of coe^Cf'e/i^ of V Peltier effect for the welding ", 6 to the quantity U2j Ta-- jg- ' which represents the quantity of heat released during the unit of time by a current of intensity equal to the unit passing from metal a to metal b. Let ©", B^ be the values of the quantity 6 inside the metals <7, 6, at a sufficient distance from the terminal surfaces. For electrical equilibrium to be established on a conductor containing 486 BOOK V. - METALLIC CONDUCTORS. the two metals a and 6, it is necessary that the potential function has, within these two metals, constant values V,^, V^, linked by the relation (£V«-h0a)-£(V/.-T-06). The quantity (i3) D^=- ^"-^^ is the potential level difference between the inner regions of the two metals a and b joined metallically. But, on the other hand, according to the definition of the quantity H [Book II, Chap. II, equality (i6)], we have _ to^n , _ of/, Equalities (12) and (i3) thus give Ut) *^" - E ^T The coefficient of V Peltier effect at the welding of two metals a, b measures the product of the absolute temperature by the derivative with regard to the temperature of the fall that experiences, in the state of equilibrium, the potential level when one passes from the inside of the metal a to the inside of the metal b. This fundamental relation was first obtained by M. Lorentz (' ). This relationship cannot be verified by experiment, at least in a direct way. Indeed, we have e^ = e(a, i, X -f-[x), e^ =e(è, i, X-t-[ji) (') H. -A. LoR^yiz, Sur l'application aux phénomènes thermo-électriques de la seconde loi de la Théorie mécanique de la chaleur (Archives néerlandaises des Sciences exactes et naturelles, t. XX, p. 129; i885). - For the history, t>oi/' P. DuHEM, Sur la relation qui lie l'effet Peltier à la différence de niveau potentiel de deux métaux au contact {Annales de Chimie et de Physique, 6° série, t. XII, p. 433; 1887). - The principles of the theory set forth in this Chapter are indicated in the latter Memoir and in P. Duhem, Le Potentiel thermodynamique et ses applications, 3* Partie (Paris, 1886). CHAP. VIII. - The PELTIEn effect. 4^7 and, consequently, equality (i4) can be written Pa = ^^'Y [Q(^> ^ X -^ (ji) - 0(a, i, X -i- [!)]. The first member of this equality can be determined experimentally; but the same cannot be said of the second, because, as we pointed out in the previous chapter, the methods for determining the potential level difference in contact with two metals determine the quantity 6(6, i, 0) - 6(a, i, o) and not the quantity e(b, i, X -r- |jl) - &(a, i, l -+- [x). Clausius (' ) had admitted that the relation between the coefficient of the Peltier effect and the potential level drop was the following He believed that this equality necessarily resulted from Thermodynamics. This relation, admitted by Clausius and by a great number of other phvsicians, must be replaced, in general, by the relation (i4)' . However, it is possible that, for certain particular metals, the relation (i4) may take the particular form ( 1 5 ) . (') R. Clausius, Ueber Amvendung der mechaiiischcn Wàrtnetlieorie auf die thermo-elektrisclien Erscheinungen ("The Mechanical Theory of Heat", translated into French by Folie, vol. - Théorie mécanique de la chaleur, translated into French by Folie, t. II). 488 BOOK V. - METALLIC CONDUCTORS. CHAPTER IX. THERMOELECTRIC CURRENTS. § 1 - Conditions in which thermoelectric currents occur. So far, we have studied systems formed by metals whose points are at the same temperature. We will now study the properties of systems formed by metallic conductors whose temperature varies from one point to another. We will assume that the shape and position of the various conductors are given, as well as the physical and chemical constitution of each of the volume elements of which they are composed, so that, in order to know completely the state of a volume element of the system, it will suffice to give the coordinates x^ y, z of a point of this element, and the absolute temperature T at this point. The internal energy U and the entropy S of the system will be given by the equalities [Book II, Chap. II, equalities (i8) and (19)] (1) EU = Er + W + Kig',-4-K2^" + ...+ K"^", {%) ES=ES +Hi^i-hH2<72-+-... + H "g'". From what we have just said, the cpiantites H and K will be continuous functions of ^, jKi -2, T, H = H(a7, jK, z, T), K= K{x,y,z,T). Let us imagine that an electric charge oq passes from the point of coordinates (x, y, z) to the point of coordinates (x -^ ox, y -i- oy, z -\- oz), the internal energy and the entropy of the system undergo variations CIIAP. IX. - THERMOELECTRIC CURRENTS. 489 ôU and 5S given by the equalities L V ^■'^ ^^ ~^ OT dx ) d\ dK dT\ (' dj ' dy ' dT dy)^^ d\ dK dK dT\ , e 1 1 ■ - ' ' dz dz dT d. -joz^dq, ss =r /dH dH dT' \dx ^^ dT dx /dH dH dT\, -^(d^ -^d-T d-^r-^ /dH dH dT\. 1, ^(dJ -^dT d^j'^^J'S' The uncompensated work involved in this change is value ox = ETSS - EoU. If we observe, moreover, that Ton has [Book II, Chap. II, equalities (16) and (17)] K = e + TH, we see that we will have dV from of dT ' tox"^ " dx ■ dT d^ dV from of dT dy ' dy ^ dT dy dV dJ-^ of dz ' of dT ■^ dT dz H^ da^ (3) ( +(^"^.-E-^^+H^)o7 H - ) 02 c^fir. dz / J ^ This equality is fundamental; it provides the laws of equilibrium and permanent motion of electricity on metallic conductors whose temperature is not the same at all points. The conditions of electrical equilibrium on a similar conductor are obtained by writing that the uncompensated work 8t, given by equality (3), is equal to o, whatever tx, ùy, 83; these conditions are thus ày ^dS dx dx Oy dy dV _^dB 'dz '^ dz of dT dT dx of dT u'^T dT dy -T-H - =0, dy of dT u^'^ dT dz -^H- =0. dz 490 BOOK V. - METALLIC CONDUCTORS. If the system has permanent currents, it follows from the fundamental hypothesis concerning permanent currents, which was obtained in Chapter IV, by generalizing Ohm's law, that we must have Sx = ( Ca; 037 -I- Cy oy -i- CzOz) dq, Cx, ^y, ^z being the components of the electromotive force at point (^, y, z). If SI is the specific resistance at the point (^, y, z), and II, V, w the components of the flux at this point, we have, according to equality (3), / to\ d& de àT "dT\ dx / , c, "i, ..., /, each of which is homogeneous, except at a very small distance (X + pi) from the surfaces that terminate it. With these metals, let's form a chain in such a way that each of them is in contact only with the one preceding it in the series and with the one following it. Let us assume that each of the contact surfaces has the same temperature at all its points, this temperature not necessarily being the same for the different contact surfaces. We will have an open thermoelectric chain. If, moreover, we suppose the two metals a. and / put in contact by a surface having the same temperature in all its points, we will have a closed thermoelectric chain. The theory, based on Thermodynamics, of thermoelectric chains, was given in 1862 by Sir W. Thomson (' ). The preceding principles easily provide the results obtained by Sir W. Thomson (2), and even allow to give them a more general form, which does not oblige to reduce the chain to a simple fd. First law. - An open thermo-electric chain cannot be the seat of any sensible permanent current. Let us consider an open chain, formed, for example, of four metals ", b, c, d {Jîg. 98), separated from each other by the surfaces S,, S2, S3, whose temperatures are T,, T2, T3. If the metal has permanent currents flowing through it, it must receive as much electricity at any given moment as it lets out, and, as these exchanges can only take place through the in (' ) Sir W. Thomson, On a mechanical theory of thermo-electric currents {Philosopliical Magazine, 4° series, t. III, p. 629; i852. - W. Thomson's Mathematical and physical papers, 1. 1, p. 3i6). (") P. DuHEM, Applications de la Thermodynamique aux phénomènes thermoélectriques et pyro-électriques. F" Part: Thermoelectric phenomena {Annales de l'École Normale supérieure, 3" série, t. II, p.4o5; i885). CHAP. IX. - THERMOELECTRIC CURRENTS. 49^ of the surface S, we see that we have (a) V [u cos,(Na,x)-^vcos{]Sa,y) H- w cos(Na,-s)] dSi=: o. The metal b must also receive at each moment as much electricity as it loses, which gives V [u cos(N;!" x) -+- pcos(N/"j') -H iv cos(N6, z)dSi -h V ["cos(N6, ce) -+- V cos{l^t "y) 4- tvcos(NA, z)] dS^- o. But equality (a) gives (?) ^ [Mcos(Ni, x) -i- V cos(N6,jk) -+- tv cos(N6, z)] dSi = o, so that the previous equality becomes (v) V [ucos{^/j,x)-h- i^cos(N6,7) -f- w cos(N/" z)] ^82 = o. We will see that, from one to the other, each surface S(, Fi g. So, S3 delivers, at each instant, a total amount of electricity equal to o. Gela posed, let us write, for the metal a, an equality analogous to equality (7), and let us transform it in the same way. Let J9a(T), ha(T), be the functions 3^(T), A(T), particular to metal a, we will have /// 3i{ii--t-v^-^w^)dxdydz = - ^[eV -{- e -}- jÇa(T)] [u cos(Na, x)^v cos(Na,jK) 4- w cos(Na, z)\ dSi -JfJ[Hix,y,z,T)-K{T)](u^^-^v^^-^".^^^^dxdjdz. 496 BOOK V. - METALLIC CONDUCTORS. If we observe that the temperature T and, consequently, the function ^"(T) have the same value at any point of the surface S, and if we take into account the equality (a), we will have /// Jl("2 -4- p2 _|_ (^,2 ) dx dy dz = - X (sV-l- 0) [U COS(Nrt, X)-^V COS(Na,JK) -4- W COS(Na, Z)] fl?Si The metals b, c, d give analogous equalities. By adding them member by member and noting that we have, at any point of S,, u cos(]Na, x)-^ V COS(Na,JK) + W cos(Na, z) -\- u co?>{'^b,x) -h V cos(N6,jK,> ■+- w cos(N/j, z) - o, and analogous equalities at any point of the surfaces So, S3, we find I f IS{.(u'^^v^--r'W^')dx,dj,dz ■fU [H(x,y,z, T) - Arf(T)] ("^-f-''j- + ^^) dxdydz. The integration in the first member extends to the entire chain. Since the second member is a very small quantity of the order of (X + {jl), the same must be true of the first; the currents can therefore have no appreciable intensity in any region of appreciable extent in the system. The electrical balance will be established on the system. We will have, at any point, according to the equality (8), H(a7, jK, -s, T) t/T = - i/(£V 4- 0). Let us take a point M {fig. 99) inside the first metal a of the chain, and a point M/ inside the last metal d of the chain. Let T and T' be their temperatures. Let us join the two ICVP. IX. - THERMOELECTRIC CURRENTS. 497 points M, M' by a line / meeting the surfaces S,, So, S3, in P, P2, P3. Equality (8) gives us .M £V(M)-f-ea(T)-sV(M')-e,/(T')= / U(x,y,z,T)dT. But the second member can be written T, I, -^T-^T-X -^'^^Jr, -^^'"A .T ^T f [H(^,r,^, T)-A^(T)]c/T+...-^ r [11(^,7, 2, T)-/irf(T)]f/T. If Ton elTects the integrations shown in the first four Fig- 99 terms, and if we neglect the following terms, which are very small quantities of the order of (-+- [jl), we find £V(M)-4-e"(T)-£V(M')-e,/(T') (9) = i5a(T,)-i5a(T)+i5^(TO-i56(T,) ( -^i5,(T3)-i^,(T2)-f-i5rf(T')-;^rf(T3). Let us assume, in particular, that the two metals a and d are identical; that the two temperatures T and T' are identical. Between the points M and M' there will be a potential level difference given by the equality l £[V(M)4-V(M')]= i5"(TO-i^6(Ti) ("o) _ -^i5/,(Tj)-i9c(T2) whereas this difference would have been equal to o, if the two metals had been joined by metallic conductors having the same temperature at all their points. D. - I. 32 (II) 498 BOOK V. - METALLIC CONDUCTORS. Second law. - A closed thermo-electric chain, in which all the welds are at the same temperature, cannot, whatever the distribution of temperatures between the welds, be traversed by sensitive permanent currents. Let a closed electric chain be formed, for example, of three metals a, ^, c {/ig- 100); let S, Sa, S3 be the surfaces which separate these three metals, and Ti, To, T3 their temperatures. Let us count as positive direction of travel the direction abcda. It is easy to show that, in time dt, each of the surfaces Si, S2, S3 is necessarily crossed in the positive direction by the same total quantity of electricity idt. If we reason as we did to establish the previous law, we can easily find -JJJ[Yl{x,y, z, T) - A.(T)] (^^. _ + ,- + ". _j clx dy dz. The last three terms of the second member are very small quantities of the order of()v+[i.); the first term of the second member disappears if we have T, = T2 = T3. CHAP. IX. - THERMOELECTRIC CURRENTS. 499 If, therefore, the solder joints of a closed thermoelectric chain are all at the same temperature, this chain cannot have a sensitive electric flux at any point in a region of sensitive extent. Third law. - V electrical balance is, in general, impossible on a thermo-electric chain where all the welds are not at the same temperature. Indeed, if the electrical equilibrium was established on a similar chain, we would have at any point (8) d{z-^T)=-}{x,y,z,T)cn. Let's draw a line that goes from metal a to metal 6, from metal b to metal c, and closes within metal a. Integrate the previous equality along this closed line and we find ^H(^,jK, 2, T)^T = o. /' Let us transform this equality by a calculation analogous to the one that gave us equality (9) and we will see that it becomes, neglecting the small quantities of the order Q^+ [J.), (12) J^a(T,)-45«(T3)-^i56(T2)-;0/,(T,)-^i5^(T3)-iPc(T2)=o. The necessary condition for electrical equilibrium to persist on a thermoelectric chain is that the temperatures of the welds verify equality (12) to the nearest quantities of V order of {\ + [t-) This condition is at the same time sufficient, because, if it is verified, the second member of equality (11) is a very small quantity of the order of Çk -\- [jl) and, consequently, the chain can have no current of appreciable magnitude in a region of appreciable extent. When the temperatures of the welds do not verify the equality (12), the thermoelectric chain becomes the seat of a permanent current whose laws we will study. Fourth law. - On a thermo-electric chain running 500 BOOK V. - METALLIC CONDUCTORS. jme by permanent currents, no flow line closes without having covered the whole chain. In any limited system, with permanent currents, any line of flow is bound to close on itself. Can it happen that a closed flux line appears, on our thermo-electric chain, as the line M PM'QM in fig. law , Fig. 101. in such a way that it is possible to open the chain with a section S that does not cut this flow line? Let us assume for a moment that this is possible. Our flux line can meet certain welds; let's imagine that it meets the weld S, at two points M, M'. Let us take this line as the director of a small flow channel of section diù. Let /"be the flux at a point, and let us pose The equalities (5) will give us Çs^fds^-fd{zS-^Q)- CH{x,y,z,T)dT, integrations extending to the closed MPM'QM line. The quantity i having the same value at all points of this line, we will have We will also have /- d{z\-he) = 0. CHAP. IX. - THERMO-ELECTRIC SHORTS. 5oi Finally we will find without difficulty J'H{x,y,z,T)dT= J [H(.r,j',z,T)-A"(T)]crr .M + / {n{x,y z,'ï)-h,{'ï)\dl. The second member of this expression is a very small of the order {\-\-[x). The same applies to i j -,-- which which shows that no line of the indicated shape can be traversed by a sensitive flow. Consider a small channel formed by flux lines. According to what we have just demonstrated, it runs through the whole thermoelectric chain as represented by ^^Jig- 102. Let's take for direction Fis;. lori. of this channel the direction abca. Let dio be its section at a point; soity the flow at this point. The quantity i =f dui has the same value all along this channel. Let us suppose, to fix the ideas, that it is possible to lead through the channel a section S, which normally meets all the flow lines once and only once. We will then easily find, with quantities of the order of Çk + ^), the equality ij^= ^a(Ta)-i5a(T0 -^J0^(T,)-i56(T2) --i5,(T2)-i5c(T3). Let due be the element that our channel cuts on the surface S. 5o2 BOOK V. - METALLIC CONDUCTORS. The quantity J = ^idQ will represent the total amount of electricity that, during the unit of time, crosses, in the positive direction, the surface S or any other section that opens the chain. As we did in Chapter IV, let us define the total resistance R of the chain for the flux distribution we consider by the equality i = S from IOC} the sign V indicating a summation that extends to all elements of the surface S, and the sign /a summation that extends to all elements of a closed flow line coming from a point of the due element. We will have, according to this definition, (i3) ^S>,{T,)-^,{T^) This is the fundamental formula obtained by Sir W. Thomson. If we call the second member of this equality the electromotive force of the chain, we can state the following law: Fifth law. - The electro-motive force of a thermo-electric chain depends exclusively on the nature of the metals which compose the chain and on the temperature of the welds. The expression (l4) C = i5a(T3)-i5a(T,) + i^6(Ti)-i56(T2)~-^,(T2)-i5.(T3) of the electromotive force of a thermoelectric chain easily provides all the properties of such chains. CHAP. IX. - THERMO-ELECTRIC CURRENTS. 5o3 § 3 - Properties of bimetallic chains. The equation (14) can be written C= i5"(T3)-i5a(T.) + i56(T,)-i^/,(T3) The terms of the first line represent the electromotive force of a thermoelectric chain formed by the two metals a and b only, having its welds at temperatures T, and T3, and counted positively when it runs the current from metal " to metal b through the weld whose temperature is T, . The terms in the second line represent the electromotive force of a thermoelectric chain formed by the two metals ^ and c only, having its welds at temperatures T2 andTs, and counted positively when it runs the current from metal b to metal c through the weld whose temperature is T2. It is thus seen that the study of a thermoelectric chain formed of several metals can always be reduced to the study of a certain number of thermoelectric chains each formed of only two metals. By means of this rule, obtained by A.-C. Becquerel ('), it is sufficient to study the properties of bimetallic thermoelectric chains. This rule has been subjected to a very precise experimental verification by Messrs Chassagny and Abraham (2). It is not necessary to study the properties of a bimetallic chain for all possible combinations of the temperatures of the two welds. Let us suppose, in fact, that we want to know the electromotive force of a chain formed by the two metals a and b whose welds are at temperatures T and To. Let us designate this (') A.-C. Becquerel, Du pouvoir thermo-électrique des métaux {Annales de Chimie et de Physique, 2" série, t. XLI, p. 353; 1839)1 {') Chassagny and Abraham, Recherches de Thermo-électricité {Comptes rendus, t. CXI, p. 4/7! 1890}. 5o4 BOOK V. - METALLIC CONDUCTORS. electromotive force, counted posilively when it tends to make the current pass from metal a to metal b through the weld whose temperature is T, by C^(T\^ Ta). We will have d:^(Ti,T2)=i5"(T2)-^"(T.) + i3/.(T,)-i96(T,.). Let To be a third arbitrary temperature. We can write i'â(Ti,T2)= i5"(T2)-i5a(To)^i56(To)-+-^/,(T2) -[i9a(Ti)-i5"(To)-i|6(To)-i5^(Tijl = d:,'i(To, T,)-tf;(To,T,). By means of this relation, due to A.-C. Becquerel ('), we see that it will be sufficient, by keeping the temperature Tq of the cold soldering fixed, to study the electromotive force of the bimetallic chain for all values T, of the temperature of the hot soldering. To study the electromotive force "L'*(To, T,), counted positively when it passes the current from metal a to metal b through the weld at temperature To, we will operate as follows. According to equality (6), we have and, consequently, equality (i4) can be written (i4') C'^{To,T,) = J[/t,(T)-A,(T)] Let us choose a standard metal, always easy to reproduce identical to itself at the same temperature, lead, for example. Let us designate this metal by the index o. Once this metal is chosen, the quantity /î"(T)-Ao(T) will be a quantity depending only on the temperature and the nature of the metal a. We will pose (i5) jK"(T) = A,(T)-Ao(T), (' ) A.-C. Becquerel, loc. cit. CHAP. IX. - THERMO-KELECTRIC CURRENTS. 5o5 and we will say that jKa(T) is, at temperature T, the thermoelectric power of metal a related to lead. The equality which gives "L'*(To, T|) will then become (.6) i:iJ(To,T,)= / [ya{'ï)-yi,{'ï)\dl. It is thus seen that if, for each metal, one knows the thermoelectric power related to lead, and this at any temperature, one knows, under all circumstances, to predict the effects of any thermoelectric chain. We shall see, in § 5, how one can determine the thermoelectric powers related to lead. For the moment, let us discuss the consequences of equality (i6). On the x-axis of a rectangular coordinate system {Jig' io3), let us plot the absolute temperatures T. Let us plot the thermoelectric powers on the ordinate. Let AA', BB' be the Fis. io3. °i_A' A r B-^ -^ curves representing the thermoelectric powers ^'"(T), j^6(T), of metals a and b. Let us lead through the points of abscissa Tq, Tj, of the parallels Tq ^o^^o" T, ^)a,, to OY. It is easy to see that the area ^o^^o^ti '^\ represents the absolute value of the electromotive force C^(To,T,), and this force will be positive or negative depending on whether the line AA' will be above or below BB'. To study the variations of the electromotive force £* (T©, ïi) with the temperature T<, it will be enough to leave invariable the line Toj^o^to and to study the variations that the area j^o^^o^il^i 1oï "S' undergoes that we move parallel to itself the ordinate T)j3,a(. It can happen that, in the whole extent of the field where the experiments are carried out, the line BB' is constantly above or constantly below the line AA': the previous area, and consequently 5o6 LlVnE V. - METALLIC CONDUCTORS. The absolute value of the torque increases as the difference between the temperature of the hot and cold weld increases. Many couples present these simple phenomena; but Fig. 104. A_ ■"^^^^ B' w t B-^ /fp. T" T^ S^ T it may happen that the two lines AA', BB' intersect at a point C i^fig- io4), whose temperature 3 is included in the field of experiments. Let us assume, to fix the ideas, that at temperatures lower than 2r the AA' curve is above the BB' curve. Let us take the temperature To lower than Sr. 1" The temperature T, starting from Tq, remains at first also lower than 2" {fig- io4). The electromotive force is then positive. The current goes from metal a to metal h through the cold probe. The electromotive force increases with the temperature of the hot solder, but slower the higher the temperature. 2" The temperature T, reaches, then exceeds the temperature 2r, F'ig. io5. which is called the/?om^ /zcM^/'e. From this moment (/""-. io5), the electromotive force stops growing, because the area Ca, (^2 must be CHAP. IX. - THERMOELECTRIC CURRENTS. 5oy counted negatively and subtracted from the Caol^Q area. However, the electromotive force remains positive at first. 3° But T, continuing to grow, can reach a value t {Jig. io6) for which the cah area is equal to the Gao I^q area. For Fig. io6. At this inversion temperature, the electromotive force cancels. When the temperature of the hot solder becomes higher than the inversion temperature, the area to be subtracted Ca, ^, becomes greater than the area Cao|3o from which it is subtracted. The electromotive force becomes negative, and the current flows from metal b to metal a through the cold solder. At the moment when the temperature of the hot solder is equal to the inversion temperature, the electrical equilibrium is established on the system, because, it is easily seen, the equality (12) is then verified. The temperature of the neutral point depends only on the nature of the two metals a and b; but it is not the same of the inversion temperature, which depends moreover on the temperature Tq of the cold junction, and is all the higher as the temperature of the cold junction is lower. If, on two axes of rectangular coordinates, the temperatures T, of the hot weld are plotted on the abscissa and the electromotive forces (To)](T,-To)f2r-I^^^'y This formula represents very exactly the results of the various experiments of Gaugain. It was first given by M. Avenarius ('). A few years later, it was found again by Mr. Tait (-). The method followed by Mr. Tait to reach it is reproduced in all the treatises of Physics. It could easily, by its form, mislead one into thinking that the formula (17) is a rigorously exact formula, whereas it is only an approximation. The very precise research of Messrs. Chassagny and Abraham (' ) shows that the differences between the results of the experiment and the formula of Messrs. Avenarius and Tait exceed the limit of the errors of observation, at least if one admits that one can confuse the temperature read on the hydrogen thermometer with the absolute temperature. If the neutral point is very far from the experimental field, the two lines AA', BB' can be assimilated to two parallel lines, and we have t^(To,T,) = fjK"(To)-jK/.(To)](T,-To). The electromotive force of the thermo-torque is proportional to the difference in the absolute temperatures of the two welds. Such a torque is called a steady-state torque. The antimony-bismuth couple has an appreciably regular operation. § 4 - Relation between thermoelectric phenomena and potential level differences at contact. The electromotive force of a thermoelectric chain formed by the two metals a and ^ and whose two welds are at the ( ' ) Avenarius , Ueber electrische Differenzen der Metalle bei verschiedenen Temperaturen {Poggendorff's Annalen, t. CXXII, p. 198; 1864 ). (") Tait, Ou tlienno-electricity {British Association Repertoriuni, vol. XLI, p. 48; 187.). (') Chassaony and Abraham, /?ec/te/-c/ies de thermo-électricité {Comptes rendus, t. CXI, p. 4:7" 602 et 782; 1890). 5lO BOOK V. - METALLIC CONDUCTORS. temperatures Tq el T) is represented by the formula "^^'(To,T,)= /[haiT)-/t/AT)]dT. Let 0a(T), 0ô(T) be the values of the quantity B at temperature T, inside the two metals a and b. It is known that [Book IV, Ch. Il, equality (i6)] /'n( 1 ) = ^j- ' "/>( 1 ) ^ J^Tp 5 which gives C^(To,T,) = [ea(T")-0/,(To)]-[0"(T,)-e/,(T,)]. But, on the other hand, when the two metals a and b are in contact, when all their points are at the same temperature T, when equilibrium is established on the conductor they form, the potential level inside conductor a exceeds the potential level inside conductor b by an amount So we have (i8) and (To,T,) = £[D;;'(T,) - D^'(To)]. This simple relation extends to a chain formed by any number of metals a, b, c, d. Let To, T, Ta, T3 be the temperatures of the welds (a, Z>), (6, c), (c, d), [d, a). We will have, by counting positively the electromotive force e when it tends to make the current run in the direction abcda (19) t=_£[D^(To)+Dg(TO + D;f(T2)+D3(T3)]. Therefore, with the factor t, the electromotive force of a thermoelectric chain is equal and of opposite sign to the sum of the FALLS that the potential level experiences when crossing the layers of thickness 2(à-|- [jl) that are close to the welds, in the direction of travel where the electromotive force is counted positively. CHAP. IX. - THERMO-ELECTRIC CURRENTS. 5ll This is the very simple relation (') which links the thermoelectric phenomena to the potential level drops which characterize the equilibrium between two metals in contact. The sign of the second member of this relation is somewhat paradoxical. Clausius(-), who was the first to look for a theoretical relationship between thermoelectric phenomena and the drop in potential level at the contact, was led to write a relationship which becomes, by using our notations (20) c = £[D;;'(To) -+- d;,;(T, ) + D^^iTo -4- d^kt,)]. According to Clausius, the electromotive force of a thermoelectric chain is equal and opposite in sign to the sum of the increases that the potential level experiences when crossing the layers adjacent to the welds, in the direction of travel where the electromotive force is counted positively. The Clausius formula would give an exact value for the thermoelectric emf, but a false sign. This remark is all the more important since a large number of authors take the relation (20) as a starting point for the theory of thermoelectric currents. The relation (19) cannot be directly controlled by experiment. The electromotive force C is measurable, but the same cannot be said for the quantities in the second member. We have, in fact, D*(To) = -^[e,(To)-e"(To)] (') P. DuHEM, Sur la relation gui lie l'effet Peltier à la différence de niveau potentiel de deux métaux au contact {Annales de Chimie et de Physique, 6* série, t. XII, p. 433; 1887). In this work I have, by mistake, considered relation (19) to be identical to relation (20). (") R. Clausius, Ueber die Anwendung der mechanischen Wàrmetheorie auf die thermoelektrischen Erscheinungeii , equation i5 {Poggendorffs Annalen der Physik und Chemie, t. XC, p. 5i3; i853. - Théorie mécanique de la chaleur, translated into French by F. Folie, t. II). 5£2 BOOK V. - METALLIC CONDUCTORS. and we have seen that the methods used to determine the potential level differences in contact with two metals only provide a measure of the quantity 6(b, i, o,To) - 6(", i.o, To). § 5 - Relation between thermoelectric phenomena and the Peltier effect. It is to Sir W. Thomson who gave, in the Memorandum we quoted at the beginning of § 2, the relation which links thermo-electric phenomena to the Peltier effect. We have seen that the Peltier effect produced during the time dt^ by a current of intensity J which passes from metal a to metal 6, was given by the formulas [Chapter VIII, equality (12)] ( d(l=-- P'/Jdt, ^ ^ P'â = g[A"(T)-A,(T)]. i" Suppose we consider the lead o and the metal a, the second equality (21) will give, by virtue of equality (10), (î2) y"(T) = -|p-(T). If, therefore, at any temperature ï, we measure the Peltier effect produced by the passage of a current from lead to metal a, we will have, by V equality (22), the thermoelectric power of metal a related to lead. M. Leroux (' ) used this method to determine the thermoelectric power of a large number of metals in relation to lead. 2" Equality (i4 bis) ,-T, Câ(To,Ti)=/ [ha{'ï)-h,{T)]dt (') F. -P. hv.Y^(iV\^ Recherches sur les courants thermo-électriques, Mémoire lu à l'Académie des Sciences le 20 avril i865 {Annales de Chimie et de Physique, 4' série, t. X, p. 201; 1867). CHAP. IX. - THERMO-ELECTRIC CURRENTS. becomes, according to the equalities ( i), 5i3 (23) This remarkable relationship lends itself to experimental verification. For example, M. Bellati(') has studied with great care the thermo-electrical electromotive force of the iron-zinc couple whose cold solder is maintained at o "C. (Tq = 2^3), while the temperature of the hot solder joint varies from o° C. to i20°C. 11 found that this electromotive force could be represented by the formula ^(To,Ti) = 9'7.77(Ti -273)- i.9488(T, - 278)2. For T, - 273= i3",8, this formula, joined to equality (23), gives P(T,) = o''", 006065. The direct measurement of the Peltier effect at this temperature gives P(Ti)=:0'=''', 005923. The error is less than^ of the number to be measured; this agreement will appear very satisfactory if Ton takes into account the numerous experimental data necessary to carry out the comparison and the difficulty that the determination of each of them presents. 3" The diagrams {Jig'' 109) which represent by the curves Fig. 109. AA', BB' the thermoelectric powers of two metals with respect to lead provide immediately, at each temperature (' ) Bellati, Atti del B. Istituto Veneto, 5* series, t. V; 1879. D. - I. 33 5l i BOOK V. - METALLIC CONDUCTORS. the Peltier refTel coefficient between these two metals. In elFet, the relations (i 5) and (21) give Pfj(T)-|[7a(T)~jK,(T)J; On inspection of fig. 109, we see that this relationship can be written P,^(T) = -laire 6a "p, Taire bartf^ being counted positively when point a is above point b. This geometrical representation allows us to predict the direction of the thermal phenomena that will occur in the vicinity of the two welds of a bimetallic chain. We will assume that the temperature varies little in the regions surrounding the two welds, so that the laws relating to the Peltier phenomenon can be applied in these regions. Let us first assume that curve AA' is constantly above curve BB' in the experimental field {^fig. io3). We know that, in this case, the current flows from metal a to metal h through the cold junction. Thus, according to the previous rule, the current heats the cold solder and cools the hot solder. Let us now consider the case where the two curves AA', BB' intersect at a neutral point C, the curve AA' being above the curve BB' when the temperature is below the neutral point. Let's assume that the temperature Tq of the cold junction is maintained below the neutral point. As long as the temperature T( of the hot solder is also below the neutral point {^fig- io4), the current heats the cold solder and cools the hot solder. When the temperature T, i^fig- io5) is between the neutral point and the inversion temperature, the current heats up both welds. When the temperature T, exceeds the inversion temperature (/fo-. 106), the current cools both welds. the THOMSON effect. 5l5 CHAPTER X. THE THOMSON EFFECT. § 1 - The electrical transport of heat. The theory of permanent currents is based, in all cases, on two hypothetical principles, one of which is discussed in Chapter IV, by generalizing Ohm's law, and the other in Chapter VI, by generalizing Joule's law. The application of the first principle to metallic conductors whose temperature is not uniform has provided us with all the consequences explained in the previous chapter. Let us now apply our second principle to these same conductors. Let us consider a metallic conductor whose points are not at the same temperature and, inside this conductor, let us trace a closed surface S. Let N/ be the normal at a point on this surface, normal directed towards the interior of the space enclosed by this surface. Suppose that the uniform currents flowing through the system are maintained inside this surface for the time dt, but that all currents outside this surface are suppressed. In this hypothesis, the system would experience a change of electrical distribution during the time dt. Each element dS of the surface S would take a charge - [k cos(N;, x)-h w cos(N/,^) 4- w cos(Ni, 5)] dS dt. This change in electrical distribution would cause a variation oU in the internal energy of the system assumed without current; this system would release a quantity of heat û?Q'= - SU. The principle that we want to apply states that the volume that limits the surface releases in time dt a quantity of heat 5i6 BOOK V. - METALLIC CONDUCTORS. which can still be written ^Q = -3U. Equality (i) of the previous chapter allows us to calculate SU and we obtain EdQ = dt^iiY -^ K)[u cos(Nj, x)-^v cos(N,-, y) -+- w cos{Ni,z)] dS. If we notice that we have [Book IV, Ghap. II, equalities (i6) and (17)] K = 0-4-TH, that we have in addition, at any point inside the surface S, of the di> dw àx dx dz ' it will be easy to see that we can write . r r ru àY d& ^c (0 dT dx ^ âl dx~^ àx)"' d\ ^ de dH ' ày'^ dj 'dy dT toy ^ d^ dy^ dy) ^ of ^ dïi ■ - -f- - -t- 1 - dz dz dz of dT dT 51 ^dH dT dT\"1 , , , T -TT, -; h li^- ]w \dx dy dz. di. dz"-2 / J"^ This equality, which must take place whatever the shape of the surface S, provides the expression of the amount of heat released in time dt by an element of volume of the conductor. If we observe that we have [Chap. IX, equality (5)] _ fdW de^ '^^T _^Y{-\ ~ \ dx dx dT dx ' dx j I dy de de_dT_"^\ ^ ~ dy dy dT dy dyj' / dT "dT ^"^^-y'Tz^Tz-^lîTz^^^y we can easily see that the amount of heat released during the CHAP. X. - the thomson effect. 5l7 time dt^ by an element of a metallic condiicleur whose temperature is not uniform and which permanent currents run through, is given by the equality Ec?Q= S^{u'^-^v'^-^Kv''-)dxdydzdt (■'-) { \ ôx '^ ôy ' " ôz ) u 'i- V - - !- w ■ - j dx dy dz dl ô^ I ffl dl dT - 'Y -=r\u \- V 'x- w -- )dx dy dz dt. dT \ ux dy dz J"^ The first term of the second member represents the heat release given by Joule's law. The second term represents the Peltier effect. It disappears if the conductor material is homogeneous around the element dx dy dz-^ but it does not change its value depending on whether the temperature distribution is uniform or not around the element dx dy dz. The third term, on the other hand, remains the same whether the conductor material is homogeneous or not around the element dxdydz] but it would vanish if the temperature were distributed in a uniform way around this element. Let/ be the electric flux at point (^, y, s). Let F be the heat flow at the same point. If K is the heat conductivity of the substance which forms the conductor, this last flow will have as components dl T. ^T . to the - K - , - K - , - K - ôx <^y dz and the third term of Ee- We can write (3) Eo(r - TS) - fl?5c=: - Crf^r, o being a quantity that depends on the nature of the reaction that the positive electricity produces when passing from M to N, on the physical state of the various bodies that take part in this reaction and on the external forces that act on the system. The variation SW can be divided into two parts; a first part 528 BOOK VI. - ELECTROLYTES. of this variation is represented by the term ^ (Y li - y a) dcj, Va and Vjj being the potential levels inside the two electrodes A and B; a second part o',W comes from the displacement that the various electrified parts of the electroljte must have undergone in relation to each other. This part could not be calculated without a complete knowledge of the laws of electrical distribution on the electroljte. However, for the reasons indicated at the end of the preceding paragraph, we cannot hope to acquire this knowledge. We shall therefore take the decision to neglect 8',W, which, as we shall see, will not prevent the results obtained from being very much in conformity with experience for batteries and voltameters of the dimensions ordinarily employed. For the same reason, we will reduce the quantity 8(61 ^Ti -4- 02 5-2-1- ...-^S "q") à (eB-eA)rf^y, neglecting the terms that can come from the change of state of the electrol}'te. The uncompensated work dz, generated by the modification we have just considered, will have the value (4) ck = [C-h{t\A-heA) - {s\n-^eK\]dq, and this modification will be impossible if we have (5) C^{tYx^ex)-{tK + YB)io. Let us now take another virtual modification: the passage of a positive electric quantity dq from point N to point M. We will have, in this case, (W H- 01^1 -+- 025-2-+-. .+ e "gr") = [(£'a-+- Oa) - (£Vb4- Vb)] rfg-. The passage of the electric charge produces some chemical reaction which may be different from that produced by transport in the opposite direction. This reaction causes E(r - TS) to vary by Eû'(r - TS); the external forces do d^^ work. CHAP. 1 - I.\ PILE. 529 We will pose (6) Eo'(V-Tl.)-d^', = -C'dq, C' being a quantity analogous to C. Our modification will give rise to an uncompensated work of z' having the value (7) dz'=[C- (t\x-i-B x)-^{zYji^ en)] dq, and this modification will be impossible if we have (8) C'- (£VA-4-eA) + (£VB+eB)^0. For there to be equilibrium in the system, electricity must not be able to flow through Teletrolyte in either direction. It is necessary, therefore, that we can find two values V;^, V^, of the new potential satisfying both conditions (5) and (8). Three cases are to be distinguished: 1" We have (9) -C' o. The inequality (y) will therefore be verified in the present case; it will be imD. - [. 34 53o BOOK VI. - MÎS ELECTROLYTES. possible to find an electrical distribution on the system which prevents the chemical reaction from occurring. 2° We have (lo) -OC. In this case, the electrical equilibrium is ensured on the system for an infinite number of values of the potential level difference between the two electrodes, In fact, for this balance to take place, it is sufficient to have -C'5(£VB+-eB)-(£VA-i-eA)â*^, or, in other words, that (Vu - V^) is at least equal to z and at most equal to -C'+eA- e" Such a system is said to be polarizable; when equilibrium is established, the system is polarized. Let's take an example. Two platinum electrodes are immersed in acidulated water. Whichever way an electric charge passes from one to the other, the water is decomposed; hydrogen is released on one electrode and oxygen on the other. The reaction being the same in both directions, we have Without electrical transport, under ordinary conditions, water cannot decompose into oxygen and hydrogen; it is in a stable state. The decomposition in question therefore corresponds to a negative uncompensated work; we have E8(r - TS) - t/5e>o, or ":: which would be established between these two electrodes if they were joined together, not by the electroljle, but by metals all having the same temperature. This property is used to obtain standard potential level differences; the Daniell stack, the Latimer-Clark stack, are constantly used for this purpose. § 3 - Properties of a closed stack. Let us now leave aside the systems of the first kind, on which the electrical equilibrium cannot be established, and let us study only the systems of the two other kinds: polarizable and non-polarizable. Given one of these systems, let us join the two metals A. and B. by a chain formed by a single metal or several metals at the same temperature and let us ask ourselves if electrical equilibrium is possible on the system thus formed. 532 BOOK VI. - ELECTROLYTES. To the equilibrium conditions already obtained, we must add the condition that we obtain by expressing that the internal thermodynamic potential does not vary when a charge dq passes from point M to point N or vice versa through the metal circuit. This condition is (t2) £VA + eA = £VB+eB. Let's see if it is compatible with the conditions found previously. i*' Non-polarizable pairs. The two conditions of equilibrium tYx-^^K - ^^h - 08+ C = O, sVA-t-eA-sVB -08 =0 are incompatible if you do not have 1^ = 0. According to the definition of C given by equality (3), this equality cannot take place unless the reaction, of which the battery is the seat, supposedly carried out without any electrical phenomenon, is a reversible phenomenon. Let us exclude this particular case and we arrive at this proposition: If Von closes a non-polarizable couple with a metal chain, V electrical balance will not be able to remain on the system. 2° Polarizable pairs. For all polarizable pairs, we have (lo) C>C. We can then classify these couples into three classes: a. Those for which we have ^. Those for which we have CHAP. I. - THE PILE. 533 Y" Those for which we have - C'>C>o. The two equilibrium conditions (îVB + eB) - (£VA-4-eA) = are only compatible for couples of the first class. Thus, among the three categories of polarizable couples, the couples of category a are the only ones on which electrical equilibrium can be established if the circuit is closed. On the others, electrical equilibrium is impossible. The polarizable pair taken as an example in § 2 falls into this category a. We will properly reserve the name of voltameters for polarizable couples of the category a. Non-polarizable pairs and polarizable pairs of the categories p and Y are such that the electrical equilibrium, possible on such a pair in open circuit, becomes impossible in closed circuit. These couples are called voltaic elements. If we close the circuit of a voltaic element, the system of which this element is part will become the seat of a permanent current. What will be the intensity of this current? To answer this question, we will make use of the first of the two hypothetical principles which dominate the whole theory of permanent currents, the principle obtained in Chap. IV of Liv. V by generalizing Ohm's law. Let's remember this principle: At a point (x, j, z) of a conductor through which permanent currents flow, the electric flux has the components u, p, w, and the specific resistance has the value ,R. If an electric charge dg passes from the point (^,J)', ^) to the point {x -+- dx, y -\- dy, z -\- dz), in the system assumed without current, it would give rise to a change that would generate uncompensated work dz and we have, whatever dx^ dy^ dz., (t3) d-z = A{u dx -{- V dy -{- w dz) dq . Now let's consider the closed circuit we want to study 534 BOOK VI. - ELECTROLYTES. dier. Let us assume that all the flow lines that are to be closed close after running through the entire chain kfië- * ' 0- Consider an infinitely untied flow channel. Let d'co Fig. III. the area of its section at a point, /the flow at this point. The product keeps the same value all along the flow channel; the flow has the same direction all along this channel. The current can, in this channel, run in the MQNP direction or in the MPNQ direction. Let us discuss these two hypotheses: i" The current flows in the MQNP direction. Suppose a positive charge dq describes an element ds of this path. In the system assumed to be currentless, it would produce uncompensated work d-z^ and we would have, according to equality (i3), dx = Si fdsdq, = -j- idsdq. doi ■' By integrating the two members along the entire closed line MQNPM, we would have rsi 01 = idq I -7- ds, J d(M 8t being the uncompensated work produced, in the system assumed without current, by the passage of a positive electricity charge dq through the closed line MQNPM. Now this work consists of the uncompensated work produced by the load dq while traversing the path MQN, work which has the value, according to equality (4), [C + sV(M) + Oa - £V(N) -+- QYi]dq CHAP. I. - THE PILE. SS") cl of the uncompensated work produced by the same load while travelling the NPM path, work which has the value [zY {N )-+- Su -t\ (M) -ex]dq. So we have oz = Cdq , cl, therefore, if the current in the considered flow channel runs in the direction MQNPM, we have (,4) C = ij^ds. 2" The current flows in the MPNQ direction. We find then, in the same way, M 5) C'=i'f-^ds. The quantities in the second members of equalities (i4) and (i5) are all positive. Each of these equalities is therefore only possible if its first member is positive. Let us see the various cases that can arise: i" Non-polarizable pairs. We then have C = - C. If C is positive, equality (i5) is impossible; in all flow lines, the current flows in the MQNPM direction. If C is negative, equality (i4) is impossible; in all flow lines, the current runs in the MPiNQM direction. '>,' Polarizable pairs. If the couple belongs to the class ^, we have C o. Equality (i4) is impossible; in all flow lines, the current runs in the MPNQM direction. If the couple belongs to the class y, we have "!^>o, C'Jfi?" = -T82, 8S being the variation that the entropy of the system, supposed in the neutral state, would undergo by the reaction produced in the system during the time dt. We also have (1-2) CJe being the work done, under the same conditions, by the external forces to which the system is subjected. Let's assume that the system is subjected only to a normal, uniform and constant pressure P. We will then have (i3) CJrff=-8[E(r-T2)-i-Pt^], V being the volume of the system. 548 BOOK VI. - ELECTROLYTES. On the other hand, we can easily find the relations ES-- A[E(r-TS)-Pp] so that the equalities (i i) and (i3) give us the relations T dC ('4) "^-"Ëdî' (,5) . ^-^idt^-^.. The first of these two relations, which assume the variables P and ï as independent variables, is due to M. H. von Helmholtz ('). This relationship leads to the following consequences: If the chemical heat in a battery is greater than the voltaic heat, the electromotive force of the battery decreases when the temperature is increased under constant pressure. If the chemical heat in a battery is less than the voltaic heat, the electromotive force of the battery increases with temperature, the pressure remaining constant. If, in a battery, the chemical heat is equal to the voltaic heat, the electromotive force of the battery, under constant pressure, remains independent of temperature. The Daniell element is in the latter case. Mr. Siegfried Czapski (2), Mr. Hans Jahn (3), Mr. Lucien Poincaré (') have verified the accuracy of the relation of Mr. H. von Helm ( ') H. VON Helmholtz, Zur Thermodynamik chemischer Vorgdnge {Sitzungsberichte der Akademie der Wissenschaflen zu Berlin, 1882, p. 2). (^) S. Czapski, Ueber die thermische Verànderlichkeit der electromotorischen Kraft galvanischer Elemente und ihrer Beziehung zur freien Energie derselben ( Wiedemann's Annalen, t. XXI, p. 209; 1884 ). ( ') H. Jahn, Ueber dieJEquivalenz von chemischer Energie und Stromenergie ( Wiedemann's Annalen, t. XXVIII, p. 491; 1886). (*) L. PoiNCARÉ, Sur les piles à ëlectroly tes fondus et sur les forcées thermoélectriques à la surface de contact d'un métal et d'un sel fondu ( Comptes rendus, t. CX, p. SSg; 1890). CHAP. II. - CHEMICAL HEAT AND VOLTAIC HEAT. 549 holtz for various batteries, containing dissolved or melted electroljtes. Let us suppose that, in a battery, the difTéi'ence between chemical heat and voltaic heat is independent of temperature; or, which amounts to the same thing, according to the formula {i4)i that the electromotive force of the battery can be represented, as a function of temperature, by an expression of the form C=KlogT + K'. In this case, the relation (i i) gives Now, if we designate by C the heat capacity under constant pressure of the system, we have The previous equality becomes 8G = o. Therefore, for the difference between chemical and voltaic heat to be independent of temperature, it is necessary and sufficient that the reaction that occurs in the system does not alter its heat capacity. If, in particular, the chemical heat is constantly equal to the voltaic heat, a case in which the electromotive force is independent of temperature, the reaction of which the system is the seat does not vary its heat capacity. This proposition is due to M. Lippmann (' ). The reciprocal of this proposition is not exact. We have, in the Daniell element, an example to which Mr. Lippmann's proposal applies. The Daniell element having, as we have seen, an electromotive force independent of temperature, the heat capacity of this element must not vary by the reaction of which it is the seat. ( ' ) Lippmann, De l'action de la chaleur sur les piles et de la loi de Kopp et Wœstyne {Comptes rendus, t. XCIX, p. Sgô; i88^). 55o BOOK VI. - THE ELECINOLYTES. In fact, the molecular specific heat of copper is approximately equal, according to Dulong and Petit's law, to the molecular specific heat of zinc. The heat capacity of a solution of copper sulfate and zinc sulfate does not change when one weight of copper sulfate is replaced by an equivalent weight of zinc sulfate. Therefore, the reactions that occur in the Daniell element do not change its heat capacity. Mr. L. Poincaré (' ) gave another example of a couple to which Mr. Lippmann's theorem applies. Let us now examine relation (i5). This relationship has the following consequences ("): If the reaction in the cell is accompanied by an increase in volume, the electromotive force of the cell decreases as the pressure increases. If, on the other hand, the reaction in the battery is accompanied by a decrease in volume, the electromotive force increases with the pressure. If the reaction is not accompanied by a significant change in volume, the electromotive force is essentially independent of the pressure. The batteries that give off gases, such as the Volta battery, are the seat of a chemical reaction accompanied by a considerable increase in volume; the electromotive force of such batteries is all the weaker as the battery operates under a more considerable pressure. On the contrary, gas cells are the seat of a chemical reaction accompanied by a great contraction; the electromotive force of these cells increases with the pressure. Finally, in batteries where no gaseous element is involved, such as the Daniell battery, the chemical reaction causes only a small change in volume; these batteries therefore have an electromotive force that is substantially independent of pressure. These are the most general properties of the flight elements (' ) L. Poincaré, loc. cit. (^) P. DuHEM, Le potentiel thermodynamique et ses applications, p. 117 Paris, 1886. CHAP. II. - CHEMICAL HEAT AND VOLTAIC HEAT. 55 1 laity. To penetrate more deeply into their study than we have done in the preceding Chapter and in this one, it would have been necessary to make an examination of the properties of dissolutions which would have led us too far. § 3 -- Two general consequences of the theory of currents permanent. The theory of permanent currents is based on two principles which we have constantly applied in the foregoing. These principles lead to certain consequences which, when generalized, will be of great use to us in the study of any current. We will end this Volume by stating these consequences. The second of the two principles we used in the study of permanent currents is the following: Let's draw a closed surface containing a part of the system. Let us imagine that the permanent currents remain what they are inside this closed surface and cancel out outside. The system would then experience, in time dt, a change that would cause the internal energy of the system to vary by 8U, assuming no current and giving rise to a diBe work of the external forces. The part of the system which is inside the surface S gives off, in time dl^ a quantity of heat û?Q given by E c?Q = - E SU -r- d(^e This equation can be applied in particular to a closed surface containing the whole system inside. It then gives the following result: The internal energy of a system crossed by permanent currents undergoes at each instant a variation equal to the variation of V energy of the system supposed without current. This proposition can be seen as a special case of the following one, which will serve as a fundamental hypothesis in the study of electrodynamics: When a system experiences a change in which 552 BOOK VI. - ELECTROLYTES. the various conductors which enclose it do not change position and in which the electric flux in each point of these conductors remains invariable, its internal energy experiences a variation equal to that which this modification would cause to the internal energy of the same system supposed without current. The first of the two principles that govern the study of permanent currents is the following: Let (a:, y-, z) and (.r -h 5^, y + 5/, z 4- oz) be two neighboring points of a system through which permanent currents flow. Let us pass a charge dq from the first to the second. This transport would produce, in the system supposed to be currentless, an uncompensated work dx, and we would have Cx, Cy, Cz being the components of the electromotive force at point £C,y, z. Let us imagine that the various material masses which constitute the system do not undergo, in the transport of the charge dq, any displacement. Let U be the internal energy of the system supposed without current and ^ its internal thermodynamic potential. We will have EU = # - T ^ , dr= - L^ and, therefore. E8U = - rfx^T ^(dz). at The transport of the charge dq from the point (x,y,z) to the point (x + 5^, y + 8/, z + 82) thus varies the internal energy of the system assumed without SU current and we have This equality takes place regardless of the load dq and the direction of displacement. Let us take in particular an infinitely untied channel formed by CIUP. H. - CHEMICAL HEAT AND VOLTAIC HEAT. 553 flow lines. Let S, S' be two normal sections of this channel. Suppose that the point (^, y, z) belongs to the first one and the point (x -f- 8x, y + OjK, z -+- 8s) to the second. Let i be the flow from the first to the second. Let dio be the area of the section S. Let ds be the distance between the two sections. Let us transport from the first to the second a quantity of electricity idiù dt. This is precisely the quantity of electricity that would be transported from one to the other if the currents remained what they are inside the small segment that the two sections cut into the flux tube, cancelling out everywhere inside this segment. We can therefore, in this case, replace ( - 8U) by the quantity of heat Chapter III. - Green's theorem i(> Chapter IV. - Gauss' Lemma. Attraction of a homogenous spherical layer 22 § 1 - The three Gauss lemmas 22 § 2 - Existence of the force of attraction at a point inside the acting loads. Consequence of the Gauss lemmas 27 § 3 Attraction exerted at a point by a homogeneous spherical layer 35 Chapter V. - Properties of the potential function at a point inside the acting charges 38 § 1 Existence and continuity of the potential function at a point inside the acting loads 38 § 2 - Existence of first order partial derivatives of the potential function. Their relation with the components of the force 43 § 3 Existence of second order partial derivatives of the function potential at a point inside the acting charges 4^ § 4 - Poisson equation 5i § 5 History 52 Chapter VI. - Criteria of the potential function of an electrified volume. Attraction of ellipsoids 57 § 1 Criteria of the potential function of an electrified volume 67 § 2 Potential function of a homogeneous ellipsoid 62 556 TABLE OF CONTENTS. VH'ftt". Chapter VII. - Electrostatic action and potential function of an electrified surface -ji § 1 - Study of the normal component of the action exerted at a point by an electrified surface 71 § 2 - Study of the tangential components of the action exerted in a point by an electrified surface 79 § 3 - Refraction of the force when passing an electrified surface 83 § 4 Potential function of an electrified surface 88 § 5 Criteria of the potential function of an electrified surface 92 § 6 - History 94 Chapter VIII. - Reminder of some notions of Mechanics 99 § 1 - The principle of virtual velocities 99 § 2 - Statement of d'Alembert's principle. Fundamental formula of the io3 dynamics § 3 - Various remarks on the connections. io5 § 4. ^ Theorem of the living forces 108 § 5 - From the potential T 1 10 § 6 - Criterion of equilibrium stability 1 14 Chapter IX. - Of the electrostatic potential 117 BOOK II. Electrical distribution on conducting bodies and the Lejeune-Dirichlet problem. Chapter I. - Condition of electrical equilibrium. Electricity is resistive at over/under conductive bodies i25 § 1 Principles of Poisson theory 126 § 2 - In the state of equilibrium, electricity resides on the surface of bodies drivers 128 Chapter II. - // there is one and only one state of electrical equilibrium i33 § 1 - Analytical translation of the Poisson principle i33 § 2 - Is there a state of electrical equilibrium? i34 § 3 - If there is an equilibrium distribution, there is only one only i36 Chapter III. - The Gauss identity and the arithmetic mean theorem , i4o § 1 . - The Gauss identity i4o § 2 The arithmetic mean theorem 142 § 3 Theorems on the variation of the potential function outside the acting loads i43 Chapter IV. - Some theorems on the sign of the electric density on the surface of a conductor i49 Chapter V. - The Lejeune-Dirichlet Problem i54 § 1 - The problem of electrical distribution can be reduced to the Lejeune-Dirichlet problem i54 TABLE OF CONTENTS. 55^ Popes. § 2 Demonstration of the Lejeune-Dirichlet principle 169 § 3 - Statement of a more general problem than that of Lejeune-Dirichlet i63 Chapter VI. - Ijx Green's function i65 § 1 - Green's problem is equivalent to Lejeune-Dirichlet's problem ,65 § 2 - Fundamental proprieties of the Green function 171 § 3 Determination of the Green's function in some cases simple , -', CuAPiTRE VIL - Transformation , in any orthogonal coordinates, of the equation AV = o. Electrical distribution on an ellipsoid . 180 § 1 - Transformation of the equation AV = o into any orthogonal coordinates 180 § 2 - Transformation of the equation AV = o into geographic and elliptical coordinates 186 § 3 - Electrical distribution on an ellipsoid removed from any influence 189 § 4- - Special cases iq', § 5 - Geometric solution of the problem of the distribution on an isolated ellipsoid i()5 § 6 - Electrical distribution on an ellipsoid subject to an influence any 200 Chapter VIII. - The method of inversion of)3 § 1 - The method of inversion or electrical images 2o3 § 2 - Application. Electrical distribution on a spherical cap . 210 § 3 - Liouville's and M. P. Painlevé's theorems 2i5 Chapter IX. - Cari Neumann's method 220 § 1 The theorem of Mr. Vito Volterra 220 § 2 The theorem of Mr. Axel Harnack 228 § 3 - Some definitions 932 § 4 Definition and properties of the fundamental function 232 § 5 Definition and properties of subordinate functions 235 § 6 - Solution of the Dirichlet problem for the interior space of a second row surface, non-biased 239 § 7 - Solution of the Dirichlet problem for the space outside a second row surface, non-biased 240 Chapter X. - The natural distribution 243 § 1 - How the solution of the Lejeune-Dirichlet problem for all bodies is reduced to the determination of the distribution natural on all bodies 243 § 2 - Method of Mr. G. Robin to determine the natural distribution on a 24 > conductor Chapter XI. - The alternating process. 202 § 1 - Extension of the Lejeune-Dirichlet problem 252 § 2 - Extension of the arithmetic mean method 254 558 TABLE OF CONTENTS. agcs. § 3 The alternating process; formation of the solution 256 § 4 - Proof of the theorem just stated 269 § 5 Successive applications of the alternating method 263 Chapter XII. - Murphy's Problem 264 § 1 - Murphy's problem ... 265 § 2 Fundamental laws of electrical condensation 269 § 3 Murphy's method 270 § 4 - The combinatorial method of Mr. Cari Neumann 2^3 BOOK III. Experimental study of electrical distribution. Chapter I. - The test body 279 § 1 - Theory of the body of test 279 § 2 - Use of the test body 283 Chapter II. - Open conductors 287 ^5 1 - Electrical distribution on an open conductor subjected to a any influence 287 § 1 - Open conductor not subject to any influence 291 Chapter III. - Level surfaces and their orthogonal trajectories 297 § 1 . - General Theorems 297 § 2 - Experimental study of level 3o3 surfaces Chapter IV. - The layers of level 3o5 § 1 . - An identity of Green 3o5 § 2 Fundamental properties of the level layers 309 § 3 - Complete study of an electrical influence case 3i2 § 4 - A special class of 3i4 capacitors § 5 - Absolute electrometer by Sir W. Thomson Thomson 3i8 Chapter V. - Green's problem and Faraday's theorems 32 1 § 1 - Green's inner problem; Green's solution 32i § 2 - Gaussian solution 323 § 2. ■ Lejeune-Dirichlet solution 328 § 4. ^ - Application to questions of electrical influence; Faraday's first theorem 329 § 5 - Green's external problem; Faraday's second theorem; electrical screens 33i BOOK IV. The internal thermodynamic potential of an electrified system. Chapter I. - Some notions of Thermodynamics 337 § 1 - Thermodynamic potential 337 § 2 - Properties of displacements without change of state 34i TABLE OF CONTENTS. SStJ Pages. Chapter II. - Determination of the internal thermodynamic potential of an electrified system 348 § 1 - How, in the study of a system, one can take into account the mutual disposition of its parts 348 § 2 - Introduction of the fundamental assumption of compatibility 355 § 3 - Introduction of the law of universal gravitation and the law of Coulomb 359 § 4 - On the continuity of quantity 8 364 BOOK V. Electrical balance and permanent currents on metallic conductors. Chapter I. - Fundamental laws of electrical balance on metallic conductors 867 § 1 - Condition of electrical equilibrium 367 § 2 - Stability of the electrical equilibrium . 371 Chapter II. - Electrical equilibrium on homogeneous conductors, laws of electrical discharge 873 § 1 - Electrical equilibrium on homogeneous conductors 878 § 2 - Theory of electric discharge; Clausius theorem 878 § 3 - Snow Harris 38i thermometer § 4 - Complete discharge of a capacitor; Riess experiments 385 55 5 Discharge of a capacitor by successive sparks 889 § 6 Cascaded batteries 892 Chapter III. - The intensity of the currents 898 § 1 - Currents flowing in the conductor's mass 898 § 2 -^ Uniform currents 4o2 § 3 - Linear currents 40-^ Chapter IV. - Ohm's law 4^8 § 1 - Statement of Ohm's law 4o8 § 2 - Statement of Ohm's law for linear currents 410 § 3 - Permanent currents in the mass of a conductor Chapter . - The permanent movement of electricity in a blade metallic 4 M" § 1 - The permanent motion of electricity in a plane blade. 4'!} § 2 - Currents in a curved blade 43 Chapter VI. - Joule's law 4^7 Chapter VII. - The potential level difference of two metals in contact 44-^ § 1 - The electrical equilibrium on a heterogeneous metallic conductor. 44'' § 2 - Some theorems on the attraction of electric layers doubles 4'^"-" 56o TABLE OF MATTER. Pages. § 3 Permanent currents in heterogeneous metallic conductors 46'. § 4 - M. Pellat's method for determining level differences potential of two metals in contact ^73 Chapter VIII. - The Peltier effect 47S § 1 - The Peltier effect 47S § 2 - Relation between the Peltier efl'et and the potential level difference of two metals in contact 4^^ Chapter IX. - Thermoelectric currents 4*^'^ § 1 - Conditions under which thermoelectric currents occur 4^8 § 2 - Properties of thermoelectric chains 494 § 3 - Properties of bimetallic chains 5o3 § 4 Relationship between thermoelectric phenomena and potential level differences at contact 509 § 5 - Relation between thermoelectric phenomena and the effect Peltier 5i2 Chapter X. - The Thomson effect 5i5 § 1 - The electrical transport of heat 5i5 § 2 - Remarks on the theory of Clausius 020 BOOK VI. Electrolytes. Chapter I. - The electromotive force of a battery 525 .1 - Faraday's law 525 § 2 - Properties of an open stack 527 § 3 - Properties of a closed stack 53i § 4 - Experimental checks 538 Chapter II. - Chemical heat and voltaic heat 542 § 1 - Distinction between chemical and voltaic heat... 5'|2 § 2 - Relation of Helmhollz 547 § 3 - Two general consequences of the theory of permanent currents 55 1 Table of Contents 555 END OF TABLE OF CONTENTS. Imprimerie GAUTHIEU-VILLARS ET FILS, quai des Gruiuls-Auguslins, i:. l \v \ >- >n t.l >ÔcA Salt Duhem, Pierre Maurice Marie Legons on electricity and magnetism PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY ^ r ) 1 \ V ^