= 00 icNJ iOO JOO -CO ijMV, (1F -YJ. \ / K s S ^" -^V > yf A \ r > LESSONS ELECTRICITY MAGNETISM l'aris. - Imprimerie C.AU l'IllKH-VILI.AKS ET Kll.S, quai des Grands-Augustins, LESSONS ON ELECTRICITY MAGNETISM, BY P. DUHEM, IN CHARGE OF A COMPLEMENTARY COURSE OF MATHEMATICAL PILYSICS AND CRYSTALLOGRAPHY AT THE FACULTY OF SCIENCES OF LILLE. TOME IL MAGNETS AND DIELECTRIC BODIES. PARIS, GAUTHIER-VILLiVRS ET FILS, PRINTERS-BOOKBINDERS FROM THE BUREAU DES LONGITUDES, OF THE ÉCOLE POLYTECHNIQUE, Quai des Grands-Augustins, 55. 1892 ( All rights reserved. ) QC LESSONS ELECTRICITY MAGNETISM. TOME II, BOOK VIL MAGNETIC FORCES. CHAPTER ONE. FIRST DEFINITIONS. § 1 - Magnetic poles ('). If two very thin magnetized needles, AB, A'B', are made to act on each other, we observe that the movement taken by the needle AB can always be attributed to four forces: two are applied to the point A and directed along the straight lines A' A, B'A; (' ) The first clear ideas on the mutual actions of magnets are due to Tobias Mayer [Theoria magnetica {Gœtting. gel. Anz., 1760)] and to ^Epinus [Examen theoriœ magneticœ a Tob. Mayero propositœ {Novi Commentarii Academiœ Petropolilanœ, t. XII, p. 3i5; 1766)]. D. - IL X 2 BOOK VIL. - THE MAGNETIC FORCES. two others are applied to point B and directed along the lines A'B, B'B. The ends of the needles to which these forces are applied are called the magnetic poles of needles. Let's suppose that A, A' are, for our two needles, the poles which, in Paris, go north, and B, B' the poles which go south. We say that the poles A, A' contain austral magnetism and the poles B, B' contain boreal magnetism. We observe that the action of A' on A and the action of B' on B are always repulsive, while the action of B' on A and the action of A' on B are always attractive; hence the law : Magnetic poles with the same name repel each other; magnetic poles with opposite names attract each other. The action that a magnetic pole. A', for example, exerts on another magnetic pole, A, for example, is equal and in the opposite direction to the action that the magnetic pole A exerts on the pole A'. The mutual actions of magnetic poles are subject to the law of equality between action and reaction. When several magnetized needles A'B', A "B", ... act simultaneously on the same needle AB, the forces that act on the latter are obtained by composing the forces that would act on it if it were subjected separately to the action of the needle A'B', then of the needle A "B", .... The action that a magnetic pole A' exerts on another magnetic pole A depends on the distance /' between the two poles; it decreases as this distance increases. Tobie Majer had already suspected that this action should vary as the inverse of the square of the distance /-. Coulomb (') and, later, Hansteen (-) demonstrated experimentally the correctness of this proposition. Their experiments leave much to be desired. We shall see, in the next chapter, how Gauss established this law with admirable precision. For the time being, we will leave the function (' ) 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 royale des Sciences pour 1785, p. 578). (' ) Hansteen, Letter to Mr. Œrsted, professor at the University of Copenhagen {Delametherie's Journal of Physics, t. LXXV, p. 4i8; 1812). CHAP. I. - FIRST DKFIMTIONS. 3 of the distance that governs the mutual action of two magnetic poles. If, on the southern pole A of a magnetized needle AB, we make act successively, at the same distance R, the poles of the same name a^ a "j ... of various needles ab, a'b', a'^b", ... and the poles of opposite name b, b', b", ... of these same needles, we will observe that the actions exerted are different. We will represent by F, F', F", ...,/, /', /", . - these actions, the repulsive actions F, F', F", ... being counted positively, and the attractive actions y, /', /"--- being counted negatively. When the southern pole A and the distance R have been chosen once and for all, these actions F, F', F", ..., f, f',f"... characterize the poles a, "', a", ..., 6, b'^ b", .... These actions are called magnetic masses or quantities of magnetic fluid contained in the poles considered. According to this definition, a southern magnetic mass is positive; a northern magnetic mass is negative. We also see from this definition that a magnetic mass pole (//i + ni!) is equivalent to the juxtaposition of a magnetic mass pole m and a magnetic mass pole m'. A magnetic mass pole o is equivalent to the absence of a magnetic pole. The experiment shows that magnetic masses of opposite signs, placed at the two poles of the same needle, are equal in absolute value. Let us take the poles ", "', "", . . ., b^ b' ^ b", ...; let us place them successively at the same distance /- from any one pole P. The pole P will undergo actions <ï>, O', 4>", . . ., o, o', o", The experiment shows that we have The actions that various magnet poles exert, at the same distance, on any pole, are proportional to the magnetic masses contained in these poles. It is easy to draw the following conclusion from this: Let P and P' be two magnetic poles; let m and m' be the masses 4 BOOK VII. - THE MAGNETIC FORCES. T is the action that these two poles exert on each other at the distance /-, this action being counted positively when it is repulsive and negatively when it is attractive. We can write T=m/n'/(/-), /(/-) being a function of the distance /', which is the same for all magnet poles, which is positive for any value of /-, which decreases when ;- increases. The function ./{?') depends on the choice of the standard pole A and the distance R; if we replace this pole and distance with another pole and distance, we increase or decrease f{r) in some ratio independent of /■. The choice of the pole A and the distance R is arbitrary; the most convenient is the following: for distance R, one takes the unit of length; one chooses the pole A, so that an identical pole A', separated from pole A by the unit of length, exerts on pole A a repulsive action equal to the unit of force. From this definition, we see that the magnetic mass of the pole A' is equal to unity. The same is true for the magnetic mass of pole A. The înagnétirjue mass of the standard pole A is V unit of magnetic mass; it is defined by this property: two magnetic masses equal to the unit, separated by a length equal to V unit, exert V one on the other a repulsive action equal to l^ unit of force. It is easy to see that this definition is equivalent to the following condition: the function f {?-) becomes equal to V unit when the distance r is taken equal to V unit of length. § 2 - Magnetic elements. Action of a magnet on a pole. The foregoing applies only to the mutual actions of magnetized needles which are very long and very thin. Here is how we can extend the considerations exposed in the previous paragraph to the study of any magnets (*). (' ) This extension is due to Coulomb [Septième Mémoire sur l'Électricité et le Magnétisme: du Magnétisme. Art. XXX {Mémoires de l'Académie des Sciences, for 178c), p. 488)]. CIIAP. I. - INITIAL DEFINITIONS. 5 Let us take a magnetized body of any shape and cut it into volume elements. We will admit that each of these elements of volume acts on a pole cl^ magnet external to V magnet as if it contained a certain magnetic mass m^ positive or austral, placed at one of its points a and a magnetic mass ( - ni) equal to the preceding one, of opposite sign and placed at a point h very close to "; in other words, we will consider the actions of this element as identical to the actions of a small magnetized needle placed along ha. The definitions and propositions given in the foregoing for magnetized needles will now easily be extended to the actions exerted by any magnet on a magnetic pole which is external to it. Let us consider a magnetic element dv or ba^ and look for its action (') on the magnetic mass M placed in A. The mass m, located in ", exerts on the mass M an action F = Mm/(/-), /- designating the distance "A and the force F being directed along aK. The mass ( - m), located at h^ exerts on the mass M an action F'= - M w /(/-'), /■' denoting the distance ^ A, and the force F' being directed along 6 A. If the pole A and the element are moved relative to each other, the preceding forces and the equal and directly opposite actions that the pole A exerts on the element perform work rfS = Ff//'+ F'c^/-'= Mm [/(r)^/- - /(/-') rf/-']. Let C5(r) be any function of r, such that we will have tZG = - MmS [o(r) - o(/'')] Let us denote by / the direction ba\ by dl the length ba; we will have dr /-' = /■ -j dl = r -{- cos ( /-. / ) dl 01 (' ) All that follows is due to Poisson [Mémoire sur la Théorie du magnétisme {Mémoires de l'Académie des Sciences, t. V, p. 347; '824)]. 6 LIVRIÎ VII. - THE MAGNETIC FORCES. el, therefore, d^ = ~M/ndlS[f{r)cos{r,l)]. The mutual actions of a magnetic element and a magnet pole admit a potential which has the value (i) P = Mmdl/{r) cos(/-, rf/). The mutual actions of any magnet and a magnet pole admit a potential, which is obtained by summing the potentials of each of the magnetic elements on the pole. Let's put (■^0 'yi)=^f(r)cos(r,l)mdl, the sign ^ indicating a summation that extends to all the magnetic elements that make up a magnet. The function X>i defined by this equality, is what we call the potential / magnetic function of the magnet at point A. The mutual potential of the magnet and the magnetic mass M, placed at point A, has then the value (3) n = Mt:>. It is seen that a magnetic element appears, in the expression of the potential function, only by the direction / and the value of the product mdl. The direction / or ba is what we call the direction of the magnetization at a point of the element. The product mdl is what we call the magnetic moment of the element. Let dv be the volume of the element, and let m dl = D]\. dv. .')1 "L is a quantity which has, at any point of the magnet, a finite value, and which we will name V intensity of magnetization at a point. It can be seen from the above that the mutual actions of a magnet and a pole outside this magnet are completely defined when the magnitude and direction of V magnetization at any point of the magnet are known. CHAP. I. - FIRST DEFINITIONS. 7 On the line /, in the direction of this line, let us carry a length equal to Oli. Let X, i)b, 3 be the projections of this quantity on three axes of rectangular coordinates O^r, Oj', Os. These three quantities Jj, al'o, G are the three components of V magnetization at a point of Vêlement dv . The knowledge of these three quantities at any point of a magnet is sufficient to define the action of this magnet on any external pole, because the intensity of the magnetization is then given at any point by the equality DfL = (a.,2 + 1)1.2-1- ©2)2 and the direction of magnetization is defined by cos(/, x) CAD cos(/, z) = 0\L These relationships allow to give a convenient form to the magnetic potential function defined by equality (2), We have, in fact, mdl = dKdv, cos(/-, /) = cos(/', x) cos(/, X) -H cos(/-,_;k) co%{l,y) -t- cos(/-, z) cos{l, z). Let x,y, z be the coordinates of point A, and ^, Tj, ^ the coordinates of a point of the element^. We have cos(r, a?) - - -- , ox cos{r, y) . - -, cos(/-, z) = dr If then, as we have already done, we define the function o (r) by the equality the magnetic potential function at point A will have the expression (5) '^{x,y,z) /[-^-*1^--^1 In the rest of this book, we will constantly encounter expressions of the form and/o being deduced from/and ^, and ^2 from ^ by rotating permutation. x = -M 5?, dx Y = --%' Z = -<■ 8 BOOK VIL. - THE MAGNETIC FORCES. To shorten the formulas, we will represent a similar expression by 11/^11 With this notation, the previous formula becomes ,,= /-|l"^^[|*. J II ^\ Il From formula (3), we see that a magnetic mass M, placed at the point {x^ jk, -s) outside the magnet, undergoes, from this magnet, an action whose components are (6) In fact, if we give the pole A a displacement whose components are Sj:, ùy^ û3, without moving the magnet, the mutual actions of the magnet and the pole will have to perform a work Xo:r H- Yoj' -4- Zos. But, according to the expression of the mutual potential of the magnet and the pole given by the formula (3), this work must also have the value -^W^'^dJ^'^-^-à^'')' which demonstrates the accuracy of the formulas (6). § 3 - Mutual potential of two magnets. Adding the potential of a magnet on a magnetic pole, we can find the potential of a magnet on another magnetic element. Let A and B be the poles of the latter; let L be the direction of its magnetic axis; let dh be the distance BA. The point A has for coordinates X, y, z, and the point B (x - ->. dLU (y ^dL), CHAP. 1 - FIRST DEFINITIONS. 9 (c - ^-,^- dLy The two poles A and B contain magnetic masses M and - iM. Let di^' be a volume element of the magnet; let (x', y', z') be a point of this element; AJ , i)l)', 3' be the components of the magnetization at this point; r be the distance from this point to point A; ;-, the distance from this point to point B. According to (3) and (5), the magnet and the pole A have a mutual potential /Il ()o(r) Il Il àx' Il Similarly, the magnet and pole B have a mutual potential ". = _,, /"Il X'^i-' He rf/. J 11 tox II But we have and .L^^L MdL^^yidv, dv being the volume of the element and DÏL its magnetization. The mutual potential of the magnet v' and the element dv has therefore the value (7) ^=dX^d.fU\x^-\^ d,'. J ûL II ôx This expression can be transformed in various ways. We have fA II j\,' ^?(^) ,1^,' ^ ^ - rlLi,' M:^ di^J dL II dx' dL dxj || ^ dx' -+- -77 - / Ma -'-V dv dL dy J II dx dz to acp(7-) ^dldlj jh^ "àx'^' ' which can still be written, according to equality (5), by designating by "Ç' (x, y, z) the potential function of the magnet v' at point (x^j^ s), rà doir) J dL dx' di>' = dx dp' dl dx - eurs, Dll~= .%, 0\l ^ dL d "^-^o, ^'^È o 10 BOOK VII. - THE MAGNETIC FORCES. from this it follows that equality (7) can be written : (8) e/v> dx We can also see, from the previous calculations, that we can write (9) l ^-'^' J [^ oxôx J y azax dx dy' dydy Once these formulas are found, we can easily obtain, by summation, the potential of the mutual actions of a magnet v and another magnet v' . According to the formula (8), this potential can be written (10) J II dx ch. It is obvious, by reason of sjmetry, that we can write it as well (10 bis) (P -I\ dx' dv'. The symmetrical form of this potential is shown in the expression obtained from formula (9) l J J \, dx ôx dx dy (") \ dy dx dy dy ■ ^.^a/Va - ; ^ - , :- C/ U") - r r - p dz dx dz dy dx dz dydz ee'p-^-'^]d.d.'. dz dz This last formula contains the laws of the forces that act between two magnets of any shape. It shows us that all these laws can be seen as the consequences of the following definition, which we shall henceforth retain to the exclusion of all others, as the definition of magnets: A magnet is defined by the knowledge of a certain geometric quantity, V magnetization, assigned to each point of its mass. CHAP. I. FIRST DEFINITIONS. The mutual actions of two rigid magnets admit a potential given by any of the three equivalent formulas (lo), (lo bis) and (i i); 'f (/-) is a function of r, the form of which will be determined later. All that precedes this definition must be considered as a simple introduction to this definition. 12) g 4 - Forces acting on a rigid magnet. A rigid magnet v is subject to the action of a number of other magnets. The actions it undergoes can always be reduced to a force (X, Y, Z) applied at a certain point O of the magnet, and to a torque (L, M, N). We can easily calculate this force and this torque when we know the magnetic potential function \'/ of the acting magnets. Let us take the point O for the origin of the coordinates and we will find, by making use of the principle of virtual velocities and of the formula (lo) Z ^ L =. M = N =■ -/( /( /( /( /( /( aAo aRo dy dx dz dx M\> dx dy cAo dx dz y ^d^ dx dz ^ dydz o ^''<'' ^~dz^ dip' di>. dv, dv, d^i^y d\o - T - -■ dx^ X d^V^ dx dy A, X dx dy dx dz dx^ at d\y ~dz dv ^■)- These formulas will allow, under all circumstances, to find the motion of a rigid magnet subjected to the action of given magnets and given foreign forces. 12 U\HE yil. - MAGNETIC FORCES. g 5. - Actions of distant magnets. Magnetic moment. We will end this Chapter with two propositions concerning the action of a very distant magnet on a magnet pole and the mutual actions of two very distant magnets. These propositions will be useful in the next chapter. The potential of a magnet on a magnet pole M is given by formulas (3) and (5). Let us assume that the magnet pole M is extremely far from the magnet. Let p be the distance from a point O (H, t), Q, inside the magnet, to the pole M. The main term of the potential will be Let's put ( 1 3 ) \= CacIv, B = CmI, dv, G = Ta dv. The previous formula can be written p = -m/(,)(a|^b| + c|). If we compare this formula with formula (i), we arrive at the following proposition: To calculate the mutual actions of a magnet and a magnet pole far away from each other, we can replace the magnet by a magnetic element located at one of its points and whose magnetic moment would be the geometric quantity (A, B, C), defined by the equalities (i3). The direction of the geometrical magnitude (A, B, C) is called A' magnetic axis of the magnet; its magnitude is the magnetic moment of the magnet. The mutual potential of two magnets is given by the equality (i i). Let us assume that these two magnets are very far apart. Let p be the distance from a point O (i, 7), ^) of the first to a point 0'(^', 'o', ^) of the second. Let's define A, B, C by the equalities (i3) and let's say A' = fx'dv\ B' = f-iiVdi^', C = fe'dv'. rnVP. I. - INITIAL DEFINITIONS. The main term of the mutual potential of the two magnets will be - BA' ^!liP-> + BB' ^44?^ + BC"^l dràl Ot] dr^' dr.dl d^ o "r/ <)^< The mutual action of two very distant magnets is the same as the action of two magnetic elements, each of which would be located inside one of these magnets and would have the same magnetic moment and the same direction of magnetic axis as this magnet. i BOOK VII. - LUS FOnCES MAGNETIQUES. CHAPTER I DETERMINATION OF THE LAW OF MAGNETIC ACTIONS AND OF THE INTENSITY OF TERRESTRIAL MAGNETISM. § 1 - Definition of the elements of terrestrial magnetism. The Earth acts on magnets; the first clear ideas on these actions are due to Coulomb (' ), who summarized the laws in the following way: The action F, which the earth exerts at a given moment on a magnetic pole of mass m, placed at a given point, is a geometric quantity given by the equality F = niT, T being a geometrical quantity entirely defined at each moment and in each point of the globe. The direction of the geometric quantityTest what is called the direction of U earth magnetic action; the value of Test V intensity of earth magnetism. The angle that the direction T makes with the horizontal plane of the place is what we call V inclination ma"-"ef/^f^e; the magnetic inclination is counted positively when the direction T pierces the horizontal plane from top to bottom 5 the magnetic inclination is positive in the regions we inhabit. (' ) Coulomb, Recherches sur la meilleure manière de fabriquer les aiguilles aimantées, de les suspendre, de s'assurer qu'elles sont dans le véritable méridien magnétique, enfin de rendre raison de leurs variations diurnes régulières {Mémoires des Savants étrangers, t. IX, p. i65; 1780). A very complete bibliography of writings on terrestrial magnetism can be found in Verdet: Leçons sur le magnétisme terrestre (Works of Verdet, vol. IV). CriAP. II. - IXTENSITK OF TERRESTRIAL MAGNETISM. 5 The half-plane formed by the vertical of the observation site and the direction of the earth's magnetic force, and the half-plane formed by the portion of the geographic meridian located north of the vertical, form a dihedral which is called the magnetic declination. If the first half-plane is east of the geographic meridian, the declination is counted positively. In the regions we live in, in our time, the declination is western, that is to say negative. Let O be an observation point. Let's conduct at this point a system of rectangular axes constituted as follows: The x-axis is tangent to the meridian of the location and directed towards the north; the y-axis is tangent to the parallel of the place and directed towards The z-axis is vertical and directed upwards. Let X,Y, Z the components, along these three axes, of the magnetic action earth. We will have, by designating by i the inclination and by o the declination, X - T cost cosô, Y - T cosf sino, Z - - T sin t. Let II be the projection of the quantity T on the horizontal plane. We will have H = Tcosf and the previous formulas can be written !X :== H cos8, Yr= Hsino, Z = - H tangt. The quantity II is what is called the horizontal component of the terrestrial magnetic action. To the preceding law, which summarizes the observations made concerning the action that the earth exerts on a magnet, we can substitute the following, which amounts to the same thing when it is only a question of calculating the forces by which the earth solicits a magnet, but which can be extended to the study of other phenomena: The earth behaves, in each point O which is exte l6 LIVIIE VII. - THE MAGNETIC FORCES. The magnetic potential function (t) of the magnet at this point O verifies the following equations: -, - - H coso, l dx ! toV " . ('2) < -.- = - H SIIIO, I -.- = H tan "f, \ az ïf , i, being three quantities which have a given value at each moment and at each point of the globe and which are called the elements of terrestrial magnetism at this point. Once these elements are known, it is easy to express the laws of the aclion of the earth on any magnet. We shall assume that this magnet is small enough in relation to the dimensions of the earth so that the quantities H, i, S can be considered as constants in the vicinity of the observation site. Then, the first three equalities (12) of the previous chapter, joined to the formulas (2), will give as components of the force applied to the magnet ^ = o, Tj = o, ^ = O. The action of the earth on any magnet is reduced to a torque. We know how Coulomb verified this proposition by experiment. The last three formulas (1 2) of the previous chapter, joined to the equalities (2), give, for the expressions of the components of this couple X = - H ( tangt / \)b dv - co%o I llî)f/cj, or, by designating by A, B, C the components of the moment CHAP. II. - IXTENSITY OF TERRESTRIAL MAGNETISM. 17 magnetic raimanl along the three axes 0.3?, Oy, Oz, [ l =- H(Btangt+ Csino), (3) I [x- H(Gcoso -t-Atangi), ( V = H(Asino - B cos8). These formulas are used to determine the three elements of terrestrial magnetism. The determination of declination and inclination is done by means of compasses; we will not study here these instruments, on which the classical treatises contain sufficient details. We will only deal with the determination of H. Let M be the magnetic moment of a magnet. The determination of H is obtained by determining successively, for this same ai M mant, the two quantities MH and '75 - We thus obtain not only the value of the horizontal intensity of terrestrial magnetism, but also the value of the magnetic moment of the magnetL We are going to expose the methods which allow to determine successively, for the same magnet, the two quantities MH M and y. § 2 - Determination of M H. Coulomb (' ) was the first to show how, for a given magnet, one could determine the product of its magnetic moment M by the horizontal component of terrestrial magnetism H; he employed two methods for this purpose: one based on the laws of torsion, the other on the laws of small oscillations. The latter has been improved by Gauss to a great extent, making it extremely precise; we will explain it in the form that Gauss gave it (-). Consider a magnet suspended from a bundle of cocoon wires in such a way that its magnetic axis is horizontal. Let's assume that this magnet is only slightly deviated from its equilibrium position and abandoned (' ) Coulomb, Septième Mémoire sur l'électricité et le magnétisme : Du magnétisme, art. II {Mémoires de l'Académie des Sciences, p. l\bb; 1789). (') Gauss, Intensitas vis magneticœ terrestris ad mensuram absolutam rcvocata {Çommentationes de Gœttingue, t. VIII; i84i. - Gauss, Werke, Bd. V, p. 81). D. - H. 2 l8 LIVRK VII. - MAGNETIC FOUNCES. to itself. It will execute, on both sides of its equilibrium position, isochronous oscillations. Let t be the duration of the oscillations, K the moment of inertia of the oscillating apparatus, S\LdM the moment of the torque which tends to bring the needle back to its equilibrium position when it is deviated from it by an angle , with respect to the same axis, a moment e(p_to_X). The pole a of the element ab contains a quantity Q of austral fluid. It exerts on the pole A of the element AB, which contains a quantity q of the same fluid, a repulsive action directed along a Pi! and having for value fyQ/(aÂ7). The pole b exerts on the same pole A' an attractive action, directed according to A!b and which has for value Point A' differs infinitely little from point A. One can consider these two forces as having approximately the same magnitude and the same THAP. II. - INTENSITY OF TERRESTRIAL MAGNETISM. 29 direction as if the point A' were in A. We then have 1. "A = 6 A ={r--i- l'-y^, 2 /being the length of the element ab and /- the distance PA. Let's put The two forces considered would have a resultant normal toAB , directed towards the east, and having the value The moment of this force with respect to the suspension axis, at the moment when its point of application has come to A', has the value 2g'Q/(p) -. Ol.cosX. The action of the element ab on the pole B is, similarly, a force normal to AB directed towards the west. If /' denotes the distance PB and if we pose this force will have the value and when the point of application of this force will come at B'; its moment with respect to the suspension axis will take the value 2^Q/(p') -, .ÔB.cosX. Let R be the distance OP; the sum of the two previous moments can be written, neglecting the infinitesimals of higher order, <7.ÂB.Q.2/^- cosX. But we have The previous moment becomes "ïm4^cosX. 3o BOOK VII. - MAGNETIC FORCES. The equilibrium condition of our AB needle is as follows: mil sin(w + X)- ()() - w - À )- mM "Aï- cosX = o. It can be written as mil sinw cosX - ()(v - co)-i- mil cosw sinX -+- OX = /"M -- r - cosX. K But the angles to and), are very small; one can replace sinw cosl by sinto by neglecting the infinitesimal of the third order, and as /n H sin a" - ("^ - w ) = o, the previous condition becomes mil cosw sinX + GX = m M ^ - - cosX. costo sin). can be replaced, neglecting the infinitesimals of the third order, by sin). 5 OX can be, to the same degree of approximation, replaced by f)sin|jL. If we pose >7l II the previous equality becomes N /(R) M ° N -H 1 R H This is the formula that gives the limit value of tang). when the two magnets are infinitely far apart. Suppose that the function /(R) can, at least for values of R greater than a certain limit, develop into a uniformly convergent series ordered along the integer and negative powers of R, (8) /(R)= A" A, . R/' R/^+i The previous equation will become ,. , > , N M Ao l.m(tangX)R = ,= ^j-^ jj- -^-. Assume the two magnets are located at a finite distance; let Vv be the distance from the figure center of magnet ab to the suspension axis. ICVP. II. - INTENSITY OF TERRESTRIAL MAGNETISM. The angle a will have a value related to R by the relation 3i <<)) tansïX N ^ M _Ao_ rT-f-T H R/^+" G Rp+3 B, C, ... being coefficients whose form does not matter. Let us now imagine a second experiment, similar to the previous one; the magnet ah is still at a great distance from the point O; its magnetic axis is still normal to AB, but, moreover, the magnet itself is placed on the normal to AB {Jlg. 5). Fi£ The needle will, in this case, be deflected eastward by an angle )/. It is easy to see that the equilibrium condition of the needle is, in this case, II - / Vx A/'N ^M /(R - /) - /(R-4-/) , wH sin(w + A )= e(p - w - X')-f- Mm '^- : cos)/, which, after simplification, can be written as follows niH cosw sinÀ'-l- sinX' = - Mm -^ - cosX', from or even tangX' = - N_ d/{R) M N~+ 1' dR H This is the formula that gives the limit towards which tang)/ tends when the two magnets are moved infinitely far apart. According to iij 3a BOOK VII. - THE MAGNETIC FORCES. pothesis expressed by the formula (8), it can be written ,. . ,,, N M Ao l.m(tangX)R=.=:j^-^ ÏÎ^rT^ If we take two magnets located at a finite distance, the angle )/ will have a value related to the distance R from the middle of the two magnets by a relationship of the form N M Ao B' G' (10) tangX =:^-^ _^^._+___ + _^^+.... In an experiment where R was always equal to or greater than four times the length of the ab magnet, Gauss found that lung). and tang // could be represented by the two formulas tangX =0.043435 1^- -f- 0.002449 -. tan g X' = o , 086 870 ^-- -i- o , 002 1 8 5 -j-:: - The comparison of these results of the experiment gives us tansX" langX/R = oo Now the comparison of formulas (9) and (10) gives /tan<ïX'\ lim ^ =/?. \langÀ/n = x The Gaussian experiment thus shows that we have (11) p = i. Thus, if we want /(R) to be developable into a uniformly convergent series, ordered according to the negative powers of R, this series will necessarily be of the form in other words, for very large distances from /-, the function f(r) varies in inverse proportion to the square of the distance. This truth, established by Gauss in a very precise manner, supports the hypothesis to which Coulomb had been led by his experiments; this hypothesis, which we will now admit, is the same as the one that Coulomb had established in his experiments, CtlAP. II. - INTENSITY OF TERRESTRIAL MAGNETISM. 33 is the following: the function /(/") is, for any value of /-, proportional to -^^- By definition, the function /"(/') must take the value i at the same time as /-; so we have M With this result, we can determine tj* The formulas (9) and (10) become, in effect, N M I B C ,, 2N M I B' C Now, by experiment, one can determine with a large approximation the coefficients of the first term of the series representing tangX and tang)/. Since N can be determined by the method we indicated in §12, we can obtain the value of the ratio jï for 1 magnet to. Knowing, for the same magnet, to determine MH and 73 , we can know the magnetic moment M of the magnet and the horizontal component H of the terrestrial magnetism. We have reduced the presentation of Gauss's method to its essential points. Many details could be added. We have assumed, in the determination of MH, that the oscillations of the magnet are infinitely small. These oscillations may have an amplitude large enough to be taken into account in the expression of their duration; this can be done by the methods indicated by Gauss (' ). It is necessary to know the direction of the magnetic axis of the magnet whose magnetic moment is being determined. Gold (-) Gauss, Anleitung zur Bestimniung der Schwingungsdauer einer Magnetnadel {Besultate aus den Beobachtungen des mq,gnetischen Vereins, 1887, p. 58. - Gauss, Werke, Bd. V, p. 374 ). D. - II. 3 34 BOOK VII. - THE MAGNETIC FORCES. schmidl (') showed how, on this magnet, one could place a mirror whose normal coincided with the magnetic axis of Faimanl. Finally, it is necessary to take into account the phenomena of magnetic induction, for the study of which we refer to the Memoirs of M. Mascart (2). (') GoLDSCHMiDT, Auszug aiis sechsjdhrigen tàglichen Beobachtungen der magnetischen Declination zu Goettingen {Ibid.) (' ) Mascart, liecherches sur le magnétisme {Annales de Chimie et de Physique, 6" série, t. XVIII, p. i; 1889). - On the measurement of the terrestrial magnetic field {Ibid., t. XIX, p. 289; 1890). CH.VP. III. - MAGNETIC POTENTIAL FUNCTION, ETC. 35 CHAPTER III. YOUR MAGNETIC POTENTIAL FUNCTION AND THE MAGNETIC POTENTIAL. !5 1. the magnetic potential function outside a magnet. We have seen [Chapiti'e I, equality (5)] that, if dv is an element of volume, of coordinates 5, yi, Î^, inside a magnet; if ri,, tHî, 3 are, at a point of this element, the components of the magnetization, we give the name toe magnetic potential function of this magnet, at a point (.a;, y^ z) which is external to it, the function do{r) ch. The 'j(r) function itself is defined as follows [Chapter I, equality (4)] do(r) dr ^-fi.r). We have admitted that we have [Chapter II, equality (12)] We can therefore take and define the magnetic potential function at a point {x,y,z) outside a magnet by the equality (') ^'^ (^, y, -) nor '^' dv. It is obvious that outside the magnet this function is 36 BOOK VII. - THE MAGNETIC FORCES. uniform, finite and continuous as well as its partial derivatives of all orders with respect to the variables a;, y, z. At infinity, it behaves like an electrostatic potential function. If, at the point (.r, y, z), there were a quantity of magnetic fluid equal to unity, this magnetic mass would undergo, from the magnet, an action whose components X, Y, Z would be given by the equalities [Chapter I, equality (6)] ^ ^ dx ^ dy Oz The expression (i) is susceptible to a very remarkable transformation. An integration allows to write it (3) 'C>(a:-,jK,z) = -^ |.l,cos(N,-,;r) __y''^"' à^ - dv , dS being an element of the surface which limits the magnet and N, the normal to this element towards the interior of the magnet. Equality (3) can be interpreted as follows: Let us imagine a fictitious fluid, which can be distributed in volumes or on surfaces in the manner of the electric fluid. Let us suppose that we distribute this fluid inside a magnet, so that it has at the point (i, tj, Q a solid density (4) P = and that we also distribute this fluid on the surface of the magnet with a surface density (5) a = - liXcos(N,-, a7)||. From formula (3), the function <){^x^y^z) will be the potential function of this dummy distribution. Let us imagine that this fictitious fluid acts on the magnetic fluids as the magnetic fluids themselves act, and the result expressed by equality (3) can be stated by saying that V magnet exerts on any external m.agnetic mass the same action as the fictitious distribution defined by equalities (4) and (5). The magnetic potential function of the magnet being identical to the ordinary potential function of this fictitious distribution, we see immediately, by the properties of the potential function CUAP. III. - MAGNETIC POTENTIAL FUNCTION, ETC. 87 that we have, at any point outside the magnet, a ((}) ^-Ç{x,y,z) = o. The magnetic potential junction is harmonic at any point outside V magnet. § 2 - The magnetic potential function inside a magnet. We have just indicated the fundamental properties of the magnetic potential function of a magnet at a point (.t:, jK, ^) outside the magnet. Let us now consider a point (.r, jk? ^) inside the magnet, and see to what extent we can extend the previous propositions to it. For points (^, r,, î^) infinitely close to the point (.r, y^ z), the quantity - becomes infinite. We can therefore ask ourselves if the integral d' dv still makes sense. Let us surround the point (i, v], 'Ç) with a small convex surface t. Either " , I J = at from the integral extends to all elements of the volume u between the surfaces t and S. The question to be examined is the following: does the integral J tcnd to a definite limit when the surface t narrows around the point (^, y, ;;), so as to return to vanish at this point by some series of shapes? The quantity - being finite for all points (^, t), Q of the volume M, one can always, by means of an integration by parts, bring the quantity J to the form CI. /TVT ^ \ d^ CI II . /^T X"^^ C I ^=^9 J = - V UIdCos(N,-, a?) 1 V Xcos(Nj, a?) / -^^ -du. r (]fter this transformation, let us contract the surface t. 38 BOOK VII. - THE MAGNETIC FORCES. The first term of J remains invariant. Let us denote by diù the angle under which, from the point {x^y, 5), oti sees the element J^ Whatever the position of the point ix^ y, z) in space, we can write the equality ■^i^ -<')/ N O H 1 /ivT N II"^^S r \\ ()A< il I 3) V(a", j,^) = - 1^ Il .l,cos(N/,;r)|| - -J II -^ I 7 dv, CHAP. III. - MAGNETIC POTENTIAL FUNCTION, ETC. 89 which we already knew to be exact in the case where the point (x, y, z) is outside the magnet. The magnetic potential function of a magnet at any point is therefore the sum of the ordinary potential functions at the same point of two fictitious distributions: one has for solid density, at any point inside the magnet, the quantity p defined by equality (4); the other has for surface density, at any point on the surface of the magnet, the quantity o- defined by equality (5). The known properties of the ordinary potential function then provide as many properties of the magnetic potential function. Let us list these properties: 1° If the quantity p has a finite value at any point inside the magnet, the function -Ç [x, y^ z) is continuous and has first order partial derivatives with respect to x, jk, ^, at any point inside the magnet. These derivatives have the expression dx - V c/the C0S(]V,-, X) (7) dz -S JId cos(N,-, x) Jlo cos(Nj, x) r dx d J II ^^ r dx di', dl /■ ,c r\\dx -r- dS- -r^ -T- di'. dz 2" If the quantity p admits at any point inside the magnet partial derivatives of the first order which are finite, the magnetic potential function V{3C^ y, z) admits, at any point inside the magnet, partial derivatives of the second order which are finite, and these derivatives verify the following condition (8) At?(a.,^,5) = 4Tr( dx dy from dz 3" The function 'Ç varies continuously, even when crossing the surface S, if the density o- is finite; but its partial derivatives undergo a sudden variation when crossing this surface; following a direction tangent to the surface S, the partial derivative of the first order of the function t? does not experience any discontinuity when crossing the surface S; but this is not so 4o BOOK VII. - THE MAGNETIC FORCES. The same applies to the derivative along the normal to the surface S. Let N,-, Ne be the two directions of the normal to a point on this surface; we have (9) ||_ + |^^=4^1i"^^cos(N,,^)|l. 4° We have assumed, in all that precedes, that the quantities ,.A.,il!>, © vary continuously from one point to another of the magnet; it may happen that these quantities are discontinuous along a certain surface; this surface will then have to carry a distribution of fictitious fluid which will have for density cr =- ||cvl>iCOs(Ni,ar)II- H ,.1,2 cos (N2, cp)\, JLf, 1)1)), G) being the values of X, il'o, G on one side of the discontinuity surface; <^.l>2, i)î)2, ©2 the values of the same quantities on the other side of the surface; N(, N2 the two directions of the normal to this surface. At the various points of this surface, we have (10) 5î^+5^2 =|Ul.,, cos(N,,^)|| + |M,2COs(N2,a7)I|. 5° If we place, at a point (^, y, z) inside the magnet, a quantity of magnetic fluid equal to the unit, the fictitious distribution defined by the equalities (4) and (5) will exert on this mass a perfectly determined action, whose components X, Y, Z will be given by the formulas Can we state this proposition, as one would be tempted to do, in the following way: A magnet exerts on a magnetic mass equal to V unit placed at an interior point (x, y, z) a perfectly determined action, whose components are given by the formulas (i i)? We will see that this expression: action of a magnet on a magnetic mass concentrated in a point inside it, can have no meaning if we admit that the law given by Coulomb and Gauss for the mutual actions of magnetic particles remains exact, even for particles in contact. CHVP. m.- MAGNETIC POTENTIAL FUNCTION, ETC. 4l The magnet is limited by a surface S; let us surround the point M(x,y,z) inside this magnet by a small convex surface a-. Let us denote by A the given magnet, whose volume is v, and by B the magnet, of volume ", included between the surfaces S and '7. Let XD [x, y, z] be the potential function at point M of magnet B; let 3, H, Z be the components of the action that this magnet B exerts on a magnetic mass equal to the unit placed at point M. Since point M does not belong to the volume occupied by magnet B, we have ùx H = Z= - ôz' According to the equalities (7), the first of these two equalities can be written -If .1,cos(N/,j:) t-^c?SWax a - Z' M 1 (1 Sal,cos(N/, 0?) ---di7-hl -Tv" I "- " ^ dx J II d^ dx of the. The quantities H and Z are likely to be expressed in a similar way. Let us suppose that we contract the surface 1 around point M so that it fades at point M. The preceding quantity varies. If it tended towards a limit whose value was independent of the series of forms through which the surface n passed to come to rest at point M, we could say that this limit represents the component parallel to Ox of the action exerted at point M by the magnet A. But we shall see that this is not the case. Let's examine separately the three terms of which S. The first term does not vary as the area rs varies. The existence of the integral already proven J II to d' ôx dv is equivalent to the fact that, when the surface t vanishes at point M, the quantity r\\ dx J lU -- of the ox 4>. LIVIIK MI. - I.KS MAGNETIC FORCES. tends to a finite limit, independent of the series of shapes through which the surface c has passed. But the same is not true for the second term of S, y-^ Il c.'l,cos(N/,a7) Il j^ ch. We will show that, if we contract the surface t in such a way that it vanishes at the point M by a series of liomothetical forms with respect to the point M, the preceding quantity tends to a given limit, but that the value of this limit depends on the initial shape of the surface o-. Let us take, in fact, one of the surfaces through which <7 passes while contracting. For this surface, we have , ■' t . i j = S a cos(Nj, x) \ ~ du -+- V {X - a) cos(N;, x) a, [i, y being the components of the magnetization at point (^, j, z). As the surface o- contracts, - tends to a finite limit; (-1, - a), (iJl) - P), (3 - y) tend to o. The second term ofy therefore tends to o. On the other hand, we have d' dx i. - X I If we then notice that, for two surfaces 1 homothetic to each other with respect to the point M, the quantities ^ cos(N,-, x) cos(/', x) - , ^ cos(Nj,jk)cos(/-, a:-)-^, V cos(N,, z)cos(/', x) - , CII.VP. in. - MAGNETIC POTENTIAL FUNCTION, ETC. yes the same value, we will see without difficulty that we have 43 limy = a V cos(N,-, x) cos{r, x) -^ -<-?§cos(N,-,7)cos(r, 07) ^ -+- Y^cos(N/, ^)cos(r, a-) ^, the three summations extending to the initial surface t. To prove what we have said, it is sufficient to show that, for two different initial forms of the surface a-, the quantity V cos(N,-, a7) cos(/', a?) -^ does not usually have the same value. Suppose that the initial shape of the surface t is that of a cylinder whose generators are parallel to the x-axis{g.6). Fig. 6. Let B and B' be the two bases. For the lateral surface, we have, in all points, cos(N/, x) = o; for the base B, we have, at any point, N/ being the exterior normal to the surface o-, cos(N,-, 3b) - i\ 44 MVnE VII. - MAGNETIC FORCES. for the base B', we have, at any point cos(N,-, x) = - I. So, for our cylinder, we have S drs r\ cos(r,x) n cos(r,x) cos{^,;x)cos{r,x)-= S^ ch- \^ ^ '-di. Let us denote by to the angle under which the base B is seen from point M, and by lo' the angle under which the base B' is seen from the same point. It is easy to see that the previous equality can be written ^ cos{^i,x)cos{r,x)^ d(s , It is clear that this quantity depends on the shape of the cylinder and the position of the point M in the cylinder. Thus the expression: action of a magnet on a magnetic mass concentrated in a point inside it has no meaning. This is a fundamental proposition in the study of magnetism; many errors have been committed by authors who have ignored it. Most of the propositions demonstrated in the above are due to Poisson (' ), although Poisson's work on these matters contains some inaccuracies (-). § 3 - Magnetic potential. From what we have seen in Chapter 1, and from the determination of the function 'f (/) obtained at the beginning of this Chapter, the potential of the mutual magnetic actions of two magnets A, A' (' ) Poisson {Bulletin de la Société Philomathique, December i8i3). - Memoir on the theory of magnetism, read at the Academy of Sciences, February 2, 1824 (Memoirs of the Academy of Sciences, years 1821 and 1822, vol. V, p. 247). (' ) See, on the subject of the inaccuracies committed by Poisson, our historical study on magnetization by influence (Annales de la Faculté des Sciences de Toulouse, t. If, 1887). CHAP. Iir. - MAGNETIC POTENTIAL FUNCTION, ETC. 45 is susceptible of two distinct expressions. It can be written [Cliap. I, equality (i i)] (,.-.) e'AO^A;) l)bx' G-.l.' c)^l dx dx' \ X\'^^' d^ dy dx' r toz dx' /- ôx ày' dydy + .1,3' 111,3' au'o' r ôzdy' à'- dx ôz r dyjz' dzdz' dv dv'. each of the integrations extending to one of the two magnets. We can also, by designating by O the magnetic potential function of magnet A, by t)' the magnetic potential function of magnet A', give to (P one of the two equivalent forms [Cliap. 1, equalities (lo) and (lo bisy\ t/x 'i 'yy il From these equalities, we deduce z' Il ^ïo' Il r II (i3) i = o-hX')'. Consider the function ^ defined by the equality ni dO r'\ ()■ - = O, - = O, -- = o. Ox Oy dz As, moreover, the function V^ is continuous in all space and equal to o at infinity, these equalities would require that we have, in all space, V = o. The magnetic potential of a magnetized system is positive, unless the magnetic potential function is equal to o in all space. The magnetic potential function will obviously be equal to o in all space if the system is not magnetized; but it is not necessary for the system to have no magnetization for the magnetic potential function to be equal to o in all space. According to equality (3) and the known properties of the ordinary potential function, it is necessary and sufficient for this to be the case that the components of the magnetization satisfy CHAP. III. - MAGNETIC POTENTIAL FUNCTION, ETC. {g at any point inside each of the magnets Tegality ()-l, dWU ()3 <)x oy oz and, at any point on the surface of each of these magnets, the equality ..l,cos(N,-, a7)-h-a)lcos(N/, j)+3cos(]N/, ^)=o. Now it is easy to find for .A,, ill), 3 different values of G which verify these equalities; the values of X, i)î), G which verify these equalities represent the components of the velocity at all points of an incompressible fluid mass which would occupy the invariant volume of the magnet; such a mass is not necessarily at rest: it is therefore not necessary that -l., ilb, S be identically zero. Formula (17) involves the quantity fd-oy /c)v'n2 /d<>Y This quantity will frequently appear in our calculations. It will therefore be convenient to adopt a single symbol to represent it. From now on, we will put With this notation ('), the equality ("7) can be written At any point outside the magnets, the quantity 11^^ has a very simple meaning; it represents the square of the absolute value of the force exerted by the magnets on a quantity of austral fluid equal to V unit placed at that point. Equality (') Lamé used, instead of the symbol WÇ, the symbol (A.V*))'. D. - II. \ 5o BOOK VII. - THE MAGNETIC FORCES. WHERE "k represents an arbitrary constant, defines in space a family of surfaces. We will give these surfaces, which we will often have to consider, the name of isodynamic surfaces. A magnetic charge, travelling through the various points of an isodynamic surface outside the magnets, is subjected by these magnets to a force whose absolute value does not vary. M 199" CH.VP. IV. - DISTRIBUTIONS EQUIVALENT TO A MAGNET. CHAPTER IV. THE FICTITIOUS DISTRIBUTIONS ARE EQUIVALENT TO A MAGNET. § 1 - The fictitious distributions equivalent to a magnet. Having a magnet, limited by a surface S, let us imagine that inside this magnet, or on the surface S itself, we distribute a certain fictitious fluid enjoying the following properties: 1° A quantity q of fictitious fluid, situated at a distance /' from a quantity M of magnetic fluid, exerts on the latter a repulsive force which has the value and vice versa. 2° Two quantities q and ^' of fictitious fluid, situated at the distance /- Tune from each other, exert on each other a repulsive action Either the ordinary potential function of this fictitious fluid. Let us suppose that we have been able, on the surface of the magnet or inside it, to distribute fictitious fluid in such a way that the ordinary potential function of the fictitious fluid is, at any point outside V magnet, equal to the magnetic potential function of the magnet. We will say that we have obtained a fictitious distribution equivalent to the magnet. The distribution defined by (4) and (5) of Chapter 52 BOOK VII. - THE MAGNETIC FORCES. The previous example gives us an example of a fictitious distribution equivalent to the magnet. This last fictitious distribution has this property, that its ordinary potential function is equal to the magnetic potential function of the magnet, not only in the field outside the magnet, but also at any point inside the magnet. It is moreover the only one, among the fictitious distributions, which possesses this last property, because it is obvious that two fictitious distributions which have, in all space, the same ordinary potential function are identical. Thus, in general, when a fictitious distribution equivalent to a magnet is found, the ordinary potential function of this distribution, identical to the potential function of the magnet at any point outside the magnet or on its surface, will differ from it at points inside the magnet. A dummy distribution equivalent to a magnet exerts the same action as V magnet on any magnetic pole outside this magnet. To this proposition, which follows immediately from the definition of a fictitious distribution equivalent to a magnet, we will add the following one, which is a little more hidden, and which we have already demonstrated in the previous Chapter for the particular fictitious distribution we studied there: The mutual actions of two magnets are the same as those of two fictitious distributions, respectively equivalent to these two magnets. Let A and A' be two magnets; let X9 and 'O' be their magnetic potential functions. The magnetic potential of these two magnets has the value, according to the equality (ly) of the previous chapter, ^=^( f'n\DdV'+- CuVdv^ ÇnX)dv'-^ Ç WO'dv' -+- fnvdv"-h fn\D'dv"^i fY-?-^-?'Adv /- Il ^ dtr || , , r II ^ ^' d"\ CHAP. IV. - DISTRIBUTIONS EQUIVALENT TO A MAGNET. 53 A" being the space outside the two magnets and dv" an element of this space. On the other hand, if we consider two fictitious distributions respectively equivalent to the two magnets A and A'; if U and U' are their ordinary potential functions, the potential of the mutual actions of the two fictitious distributions will be Y=-/-/ fui]dv-\- CnUdi'-^ fuVdv'^ fai]'di>' ''^ VA -'a -V -^A' r r r\'di] dU' '\ r II dV dV II ^ , r II '^u dij' |i , " J , I OX OX II J K"\\ "-^ '^'^ I' It is necessary to prove that if we move the two magnets, one with respect to the other, without altering their shape or their magnetization, we will have 8iT = oY. Now, in the first place, we have, in the spaces A' and A", 1') = U, and, in the spaces A and A", t')'=U'. Second, the terms fuvdv, f nu di>, fnXD'di', fnu'dv', ^K ''a"^A' '',V are invariant in therefore, any reduction made, remain invariable in the modification of which it is about. We have dV \ dVYU ~dx ) dx dv r II fd-o' dV\ dx) Il , ;i Jy II \''^ OX J ÔX \\ J Green's theorem allows us to transform the quantity between brackets into + f {U - ■0)àV'dv-h f {{]' - V ) W dv' . 54 BOOK VII. - THE MAGNETIC FORCES. But, at any point of the surface S, we have U - t') = o ; at any point on the surface S', we have U'- iT^o; any point of magnet A being external to magnet A', we have, at any point of magnet A, At)'=o; any point of magnet A' being external to magnet A, we have, at any point of magnet A', At') = o. So we have J - SY = o, which demonstrates the stated proposition. Given a magnet, one can find an infinite number of fictitious distributions equivalent to that magnet. All these distributions have this property: The total amount of fictij fluid that forms a distribution equivalent to a given magnet is the same for all distributions equivalent to this magnet. This proposition can be demonstrated quite easily in the following way: Let us consider a magnet bounded by a surface S and a distribution equivalent to this magnet. This distribution, which contains a quantity Q of fictitious fluid, has an ordinary potential function U. Let us surround the magnet with a closed surface S' {flg- 7). Let N^ be the exterior normal to this closed surface. According to the lemmas of GausS; we have But, at any point outside the magnet, the ordinary potential function of the fictitious distribution is identical to the magnetic potential function t) of the magnet. We have therefore CHAP. IV. - DISTRIBUTIONS EQUIVALENT TO A MAGNET. 55 This equality shows that, according to the stated proposition, the quantity Q is determined when the magnet is known; it is the same for all fictitious distributions-equivalent to the same magnet. Fig. 7. S' To calculate the quantity Q, we can, according to the previous theorem, take any fictitious distribution equivalent to this magnet, for example the fictitious distribution studied in the previous chapter, which has, at any point inside the magnet, a solid density and, at any point on the surface of the magnet, a surface density (2) a=:-||.^cos(N,-,a7)lI. For this distribution, we have The first of the two terms that make up Q can be integrated immediately; it is found to be equal to and of opposite sign to the second; we therefore have Q = o. Hence the following proposition: Any fictitious distribution equivalent to a magnet contains as much positive fictitious fluid as negative fictitious fluid . 56 VRE VI! - MAGNETIC FORCES. § 2 The surface distribution equivalent to a magnet and the derived Lejeune-Dirichlet problem The particular fictitious disiribulion defined by equalities (i) and (2) has fictitious fluid distributed inside the magnet and fictitious fluid distributed on its surface. In some cases, the latter distribution exists alone. This is what happens, for example, if the magnetization is uniform. The magnetization of a body is said to be uniform when the intensity of the magnetization has the same magnitude and the same direction at all points of the magnet. The three quantities al., i)\), S have then independent values of .r, j', 5, so that we have d.% _ d\s\^ _ dZ _ Ox ^ Oy ^ àz and, therefore, p = o. In this case, the equivalent surface distribution of the magnet has density ff = Il r,il,cos(N/, x)\. It is easy to obtain a geometrical representation. Let us give the surface S which limits the magnet an infinitely small translation parallel to the uniform magnetization (-1., iiï), S) of the magnet. Let S' be the new position of the surface S. The surface density T at a point on the surface S will be proportional to the distance from this point to the surface S', this distance being counted positively when in the vicinity of the point considered the surface S' is outside the surface S. In a more general way, the surface distribution exists alone whenever -l>, ill), 3 vary inside the magnet, so that we have at any point dX ^^^ dB -- -h -. H -V- = O. ox oy ôz Such a magnet is what Sir W. Thomson ( ' ) calls a magnet (' ) See Chapter VU, § 1. CHAP. IV. - KyUlVALEXTKS DISTRIBITIONS \ LX AIMANT. 67 solenoidal; when we study, in Book VIII, the theory, given by Poisson, of magnetization by influence, we will see how important is the study of solenoidal magnets. In the case where the magnet is solenoidal, its external action is the same as that of a purely superficial fictitious distribution, whose density is given by equality (2). In all possible cases, one can find in one way and in one way only a fictitious distribution, entirely spread on the surface of a magnet and equivalent to this magnet. Only in the case where the magnet is not solenoidal, the surface density of this fictitious layer no longer has the value given by equality (2). The demonstration of this theorem follows immediately from the principles laid down in Book III, Chapter V, §§ 1, 2 and 3. We have seen, in fact, that Ton could, in one and only one way, distribute a given quantity of fluid (here equal to o) over a surface S, in such a way that the potential function of this fluid is identical, outside the surface S, to a given harmonic function, which is here the magnetic potential function t;; of the magnet. Given a magnet, how will one determine this surface distribution equivalent to it? The answer to this question depends on how the magnet is given. If the magnetization is given at each point of the magnet, it will be possible to calculate the value of the magnetic potential function at any point of the space outside the magnet or of its surface; it will then suffice to solve Dirichlet's problem for the space inside the magnet, and the equivalent surface distribution will be known, in accordance with the principles that were explained in Book III, Chapter V. We will operate in the same way if, instead of giving ourselves the magnetization at each point of the magnet, we give ourselves directly the magnetic potential function in all the space outside this magnet. If we only take the value of the magnetic potential function at the various points on the surface of the magnet, we will have to solve the Dirichlet problem for the space outside the magnet and for the space inside the magnet in order to determine the surface distribution equivalent to it. 58 BOOK VII. - THE MAGNETIC FORCES. It is not in any of these forms that the problem is posed to a physicist who is given a real magnet and asked to determine the surface distribution equivalent to this magnet. This physicist has no means of determining the magnitude and direction of the magnetization at each point inside the magnet; nor does he have any means of determining the value of the magnetic potential function at the various points of the magnetic field or the surface of the magnet. All he can determine, by methods that we will study in the next paragraph, are the X, Y, Z components of the action that the magnet would exert on a magnetic pole equal to the unit placed at a point [x, y, z] outside the magnet and not too far from the magnet. We know that these components are related to the magnetic potential function by the relations V à-Ç à-Ç d-Ç ox Oy oz We see that the experiment only allows us to determine the partial derivatives of the magnetic potential function at the various points of the field. From then on, this is the form in which the problem of determining the equivalent surface magnetic distribution of a magnet will appear to the physicist. Having determined by V experiment the value that takes, at the various points outside V magnet and infinitely close to its surface S, the derivative -^rr of the magnetic potential function following the normal outside V magnet, find the fictitious surface distribution that is equivalent to this magnet. Let U be the potential function of the sought fictitious distribution. Let us consider the space outside the magnet. In this space, the function U is harmonic; at infinity, it behaves like a potential function; on the surface S, -rrp takes values don We know (Book II, Chap. V, § 3) that there is only one function U satisfying these conditions. To determine this function is to solve, for the space outside the surface S, the problem CHAP. IV. - DISTRIBUTIONS EQUIVALENT TO A MAGNET. 5ç) to which we have given the name to the problem derived from LejeuneDirichlet. Once this problem is solved, we will know the values u that U takes on the surface S. Let us consider the space inside the surface S. The function U is harmonic in this space and it takes, on the surface S, given values u. There is only one function U which satisfies these conditions; it will be obtained by solving, for the space inside the surface S, the Lejeune-Dirichlet problem. The function U being then known in all the space, the density t of the fictitious fluid in any point of the surface S will be obtained by the formula (3) i. -h 1 471 -+ Thus, the determination of the equivalent surface distribution at a physically given magnet requires that V one first solve the derived Lejeune-Diriclilet problem for V space outside of V given magnet, and then the Lejeune-Dirichlet problem for the space inside that magnet. Some remarks about the previous solution: I** This solution involves a check. Once the density , or, which amounts to the same thing, the function U, for the space outside the magnet. In this case, it is sufficient to solve the Lejeune-Diriclilet derivative problem for ()o LivnE vir. - Magnetic fokcks. the space outside the magnet; it is unnecessary to solve the Lejeune-Dirichlet problem for the space inside the magnet. 3° For a very long time, physicists, and in particular Jamin, had the most erroneous ideas concerning the determination of the fictitious layer equivalent to a magnet. They simply took, as an expression of the density of this fictitious layer, - ,_ i_ ât^ This expression could only be exact, as can be seen by comparing it to equality (3), if we had, at any point on the surface of the magnet The function U would then be harmonic inside the surface S and would verify equality (4) at all points of the surface S. From what we have seen (Book II, Chap.V, § 3), all the functions U which satisfy these conditions inside the surface S differ from each other, inside this surface, only by one constant. As the function U = o satisfies these conditions, we see that any function which satisfies these conditions is constant inside the surface S and on the surface S itself. The fictitious layer would be in equilibrium by itself on the surface S. The fictitious layer contains as much positive fictitious fluid as negative fictitious fluid; we know that such a layer could not be in equilibrium by itself on the surface S without its density being equal to o at every point of this surface. Thus the hypothesis admitted by Jamin for the determination of the fictitious layer equivalent to a magnet would be exact only if the density of the fictitious layer were, at any point, equal to o, in which case the magnet would have no action on the external points. § 3 - Experimental methods for the study of the fictitious distribution. We have seen that, in order to determine analytically the fictitious surface distribution equivalent to a magnet, it was necessary, first of all, to determine the experience CHAP. IV, - DISTRIBUTIONS EQUIVALENT TO A MAGNET. 6i mentally the value of -r^ at the various points of the magnet. The The methods used for this determination can be classified into two types: i" Coulomb's method; vi" The Van Rees method. Another method, known as the V-tear method, has been proposed by Jamin (' ), and used by this physicist and by Mr. Duter. We will see later (Book IX, Chap. IX, § 4) .11 - , 1 , . d^) that this method does not serve to determine t^t*. i" Coulomb's method ('-). - Suppose a thin, very long, magnetized needle is suspended from the cocoon wire of the torsion balance. It is in equilibrium under the aclion of the earth in a certain horizontal position AB {^fig. 8). The torsion of the wire has an unknown value. Fi g. 8. We approach the magnet for which we want to determine the value d4--a=o. We will thus obtain the component, along the normal to the surface of the magnet, of the action that the magnet exerts on a mass CHAP. IV. - DISTRIBUTIONS EQUIVALENT TO A MAGNET. 63 equal to the unit placed at point A; but, in defining this action, we must not forget that the distribution on the magnet is that which occurs in the presence of the needle AB. We will admit that V approaching V needle AB does not appreciably modify the magnetic distribution on the magnet studied. With this new approximation, the force Y^ will be the one produced by the magnetic distribution that we want to study on a magnet pole equal to the unilevel infinitely close to the surface of the magnet. We will thus have T^ having the same meaning as in the reasoning that above. Thus, by means of approximations whose degree of accuracy is rather difficult to appreciate, Coulomb's method provides the experimental data that it is necessary to know in order to study the fictitious distribution of magnetism. Coulomb also employed, to measure the force Fi^, instead of the torsion balance, a method based on the duration of the oscillations of a small needle in the presence of the magnet. The method can be justified by about the same considerations as the previous one; it is subject to the same approximations. 2° Van Rees' method, modified by Mascart and Joubert. - Van Rees was the first to have recourse to the phenomena of electromagnetic induction to study the distribution of magnetism. 11 employed this method, as we shall see in the next chapter, in the examination of linear magnets. MM. Mascart and Joubert (' ) have shown how this method can be modified, so as to make use of it for the study of the magnetic distribution on any magnet. Let's take a very small, flat, closed circuit of area Q. Let's move it in the magnetic field of a magnet. Let N be the normal to the positive side of this small circuit. At the instant <, this displacement generates in the small circuit an integral electromotive force (' ) Mascart and Joubert, Leçons sur l'électricité et le magnétisme, t. II, p. 728; Paris, 1886. 64 LlVRIi: VII. - LKS FORCKS MAGNETIC. of induction C, which has the value (') [Book XV, Chap. III, equality(7)], dN \Ti dt d^ expression which is exact, either that the magnetization of the magnet varied during this displacement, or that it did not vary. This being the case, let us connect the small circuit by a double wire to a ballistic galvanometer (Book XII ï, Chap. Vil) placed very far from the magnet. Let us initially place this small circuit very close to the magnet (Fig. 9). of the magnet, so that its plane is parallel to the tangent plane at M to the surface of the magnet and the normal N to its positive face coincides with the normal Ng. No current flows through the wire; the magnet is therefore in its natural state; the value initial ( -rir of -r^, is thus well the quantity -tt- cf we want determine. Let's quickly remove the small circuit to stop it only at an extremely large distance from the magnet; the final value dV\ from dV ,^, , v^v. ^iTr is extremely small. The small circuit is in the neutral state at the beginning of the experiment and at the end. Let E be the electroraotor force that would be present in this circuit at time f, if the current flowing through it were uniform. Let R (') tj is the fundamental constant of electromagnetic actions. CHAP. IV. - DISTRIBUTIONS EQUIVALENT TO A MAGNET. C) the total resistance of the system of which it is a part. It follows from the considerations that will be explained in Book XIV, Chapter VII, that the total quantity Q of electricity set in motion in our small circuit can be calculated by the formula But we have ^-J E^C Edt. C' being the integral electromotive force that the circuit would induce on itself at time t, if the current through it were at all times uniform. The circuit being undeformable, the current through it being equal to o, at the start and at the end, we have / C'd/ = o. 11 thus remains or, according to the above, Since the ballistic galvanometer allows us to know Q, we can know -.-^ by a method that is no longer subject to the uncertain approximations of the Coulomb method. dN ,] D. - II. 66 BOOK VII. - THE MAGNETIC FORCES. CHAPTER Y. THE PROBLEM DERIVED FROM LEJEUNE-DIRICHLET. § 1 - The Lejeune-Dirichlet derived problem for cylinders. We have seen that the determination, from experimental data, of the fictitious magnetic distribution equivalent to a full magnet leads to the solution, for the space outside this magnet, of the Lejeune-Dirichlet problem. This problem can be stated as follows: Letp be the surface of V magnet; let N^ be the external normal to this surface; we ask to find a function x? harmonic in all the space outside the surface S, behaving at V infinity as a potential function and such that -^ takes on the suif ace S of finite, variable data values in a continuous way. It would be desirable to have methods for solving this problem as powerful as those created for solving the Dirichlet problem itself. Unfortunately, this is not the case, and we can only solve the problem in question in a few rather special cases. Let us suppose, first, that the magnet has the shape of a practically very long and theoretically unlimited cylinder, whose generatrices are parallel to the 5-axis, and let us admit that the magnetization of this magnet is the same at any point of a line parallel to the z-axis. The magnetic potential function "Ç will then be independent of z. Let L be the contour of the section of the cylinder through the XOY plane. CHAP. V. - THE DERIVED PROBLEM OF LEJEUNE-DIRICHLET. 67 Let Nff be the normal to the curve L outside the area bounded by this curve. The proposed problem will then be reduced to this one: Find a function t? of the two variables x and y, harmonic at any point of the unlimited plane area outside the curve L, behaving at V infinity as a potential function and such that -y^ p retains, at any point of the curve L, given , finite, continuously variable values. This problem is nothing else than the derived Lejeune-Dirichlet problem, reduced to the case of two variables. Now, in this case, if we know how to solve the Lejeune-Dir ichlet problem for unlimited area outside the curve L, we know, for the same area, how to solve the derived Leje u n e-Dir ich let problem. Let us start by determining a function v{x^y)^ harmonic at any point of the unlimited area outside the curve L, behaving at linfîni as a potential function, and taking at any point of the curve L a constant and positive value given a. We know how to do this operation since, by hypothesis, we know how to solve the Lejeune-Dirichlet problem for the region outside the curve L. Consider the two families of curves V = const., u = const, the curves of the second family being the orthogonal trajectories of the curves of the first family. These two families of curves form an orthogonal curvilinear coordinate system. In this system, the linear element is represented by the expression A and B being two positive functions of u and v. According to what has been shown elsewhere (Book V, Chap. V, § 2), the orthogonal coordinate system in question forms an isothermal system, so that we have k = Y{u,v)f{u), B = ¥{u,v)g{v), F(m, p), /(u), g{v) being three positive functions. 38 BOOK VII. - THE MAGNETIC FORCES. According to the principles explained elsewhere (Book II, Ghap. VII, § 1), the equation will become, in the new coordinate system, A 'dv ) d /A (Jt?\"^ /B d-Cn well Let's take two new variables a, [îi, linked respectively to u and t^ by the relations "^0 dv. The function F("/, i') will become (a, [i). The square of linear Telement will have the new expression (i) rf52=^4>2(a, p)(Ja2+rfj32). The equation which expresses that the function \'^ is harmonic will become simply ()2-C> <32-C) Moreover, the two families of coordinated lines a = const, p = const. will coincide respectively with the line families u = const, V = const If we put K= f f{u)du, the line L will be represented by the equation a = K. With these preliminaries in mind, we will first note that. I CHAP V. - THE PROBLEM DERIVED FROM LEJEUNE-DIRICHLET. 69 if the function "Ç verifies the eqviation (2), the function XD(a:,y) defined by will also verify this same equation. This function is therefore harmonic in the region outside the curve L. We will notice in the second place that we have, at any point of the curve L, We ~ <Ï>(K, P) "di or The function XD thus takes, at any point of the curve L, values which can be regarded as given. The function XD will then be determined by solving the Lejeune-Dirichlet problem for the region outside the line L. Once the function VJ is known, we will obtain the value of the function t;> at a point with coordinates a, (3), by the formula -(?(", P)- f 0(a, |3)c/a, so that a quadrature will complete the solution of the problem. § 2 The Lejeune-Dirichlet derivative problem for the sphere. In the case where there are only two variables in the question, the derived Lejeune-Dirichlet problem is reduced, as we have just seen, to the Lejeune-Dirichlet problem. This reduction relies essentially on the possibility of making the contour of the studied area appear in the number of lines that compose a system dividing the plane into infinitely small squares. A similar reduction would take place in space if the surface of the magnet could be part of a triply orthogonal system dividing space into infinitely small cubes. In what case will the surface of the magnet possess a similar property? 70 BOOK VII. - THE MAGNETIC FORCES. The classical system of rectangular coordinates forms a system dividing space into infinitely small cubes; this system corresponds to the case where the magnet has the shape of a theoretically unlimited plate, enclosed between two parallel planes. In this case, if we suppose that one of the faces of the plate is formed by the XOY plane, the magnet being located below this plane, we will start by determining a function '0(x, y, z), harmonic in all the space located above the XOY plane, behaving at infinity as a potential function, and taking at the various points of the XOY plane values equal to the values of -j- given by the experiment. Once this is done, the function 'Ç will be determined at any point of the space considered by the formula \9 = - / t) dz. -l Can we find another triply orthogonal system dividing space into infinitely small cubes? If we notice that such a system would be a conformal representation of the first one, and if we remember the theorem demonstrated by Liouville (Book II, Chap. VIII, § 3), we see that such a system must be deduced from the previous one by inversion. Since a plane is transformed into a sphere by inversion, we come to the conclusion that the method by which, in the case of two variables, one reduces the derived Dirichlet problem to the Dirichlet problem is applicable to the space outside a sphere, but not to any other form of unlimited space. As we know how to solve the Dirichlet problem for the space outside a sphere, we will also know how to solve the derived Lejeune-Dirichlet problem for this space. Here is the simplest and most practical way to actually solve this problem: Let us mark the situation of a point outside the sphere by its distance r from the center of the sphere, its western longitude t]> and its northern colatitude 8. The function *= ,:,Y,(0,4.)-+-^Y,(e,<|.)--...-,-^;;^^Y"(e,4.)-i-..., Y"(0, (|>) being a Laplace function which depends on ('^/i + i) coefficients. It is these coefficients that we have to determine. Now, from the previous equality, we deduce di l=-'r [^ Y"(^' ^^-^ ^^^(^' -!')+- --^ 7.^ Y"(0, 4^) + ...] On the other hand, at the surface of the sphere, - takes the given values -r^- If therefore we designate by R the radius of the sphere, we must have, at any point of the sphere equality which will allow easily, according to the properties of the functions Yrt, to determine the coefficients of these functions, and, consequently, to determine "Ç. It is by this method that Gauss ( * ) was able to give the series development which represents the magnetic potential function of the globe at any point outside the Earth or located on its surface, taking as a starting point the determination, at each point on the surface of the globe, of the vertical component of terrestrial magnetism. This development is of great interest. Indeed, if we designate at a point of the globe by X the horizontal component of the terrestrial magnetism directed towards the geographical north, by Y the horizontal component directed towards the east, by Z the vertical component (' ) Gauss, Allgemeine Théorie des Erdmagnetismus (Gauss and Weber, Resultate aus den Beobachtungen des magnetischen Vereins im Jahre i838. Leipzig, 1889. - Gauss, Werke, Bd. V, p. 119). 7'2 BOOK VII. - THE MAGNETIC FORCES. directed towards the zenilii, we will obviously have ^ ■' j RsinÔ 4' Once we have obtained, by the method of Gauss, the development^of "Ç for any point outside the Earth or located on its surface, the equalities (3) allow us to calculate the elements of terrestrial magnetism at all points on the surface of the globe. Two remarks about the determination of the magnetic potential function of the Earth: i" The method we have just indicated assumes that the value of Z or of -r- has been determined experimentally at any point on the surface of the globe, but in reality this determination can only be made at a limited number of stations. It would therefore be impossible to determine all the coefficients which appear in the development of 'Ç in an unlimited series. We will therefore be satisfied with representing approximately t? by the limited series which depends on 3-t-5+...-l-(2 "H-l) = (/î-Hl)2- I unknown coefficients; [('? + -)" - determinations will suffice to make their value known. 2° The experiment does not directly determine the value of the vertical component Z of the terrestrial magnetism, but the values of the horizontal component H and the inclination i. From these values, we deduce the value of Z by the formula Z = - tangi. The Gaussian method gives H with great precision; but CHAP V. - THE DERIVED PROBLEM OF LEJEUNE-DIRICHLET. 78 Since inclination compasses are the least perfect of all magnetic instruments, Gauss was concerned with finding an accurate method for determining magnetic inclination. He succeeded in doing so by employing the properties of earth-induced currents, as we shall see in Book XV, Chap. III, § 2. 74 BOOK VII, - MAGNETIC FORCES CHAPTER YI. LINEAR MAGNETS. In this chapter we will study in a special way the magnets whose two dimensions are extremely small compared to the third. We will call these magnets "linear magnets. Consider a magnet whose points differ very little from a line s or BA {fig- lo). On this line, consider two Fig. 10. points M, M', infinitely close, whose ds is the distance. Through these points M and M' let us lead straight sections in the magnet. Let w be the area of one of these straight sections. The volume of the element MM' will be iùds. If 3Kj is the magnetization intensity at a point of this element, its magnetic moment will have the value DW-iùds. We pose [jt = DlLu). Consider a point {ce, y, z) located at a distance r from the elevation CHAP. VI. - LINEAR MAGNETS. jS ment MM'. The magnetic potential function of the element MM' at this point (x, y, z) will have the value [Chapter I, equality (i)] cos(/-, /) y = lias; - > l being the direction of the magnetization of the MM' element. We will assume that the linear magnet under study is magnetized longitudinally, i.e. that the direction / of the magnetization coincides at each point M with the direction of the tangent to line 5. We will then have cos(r,/) = -^ and V=u. -f ds. bones The value at the point (^, y, s) of the magnetic potential function of the whole magnet is the sum of the quantities P relative to the different elements MM' of the magnet. We have therefore, by designating by L the length of the magnet, / dl An integration by parts allows to transform this equality into (I) 'Ç(cr,y,z) = ^-^- ^-fds, ri Tq J^ r ds , . jjlq and [X, being the values of [jl at points B and A, /'o and r, being the distances from points B and A to point (ûc, jv', z). It can be seen from this formula (i) that the action of a linear magnet on an external point can be replaced by the action of a fictitious distribution formed by : 1° Two masses of fictitious fluid, respectively equal to - {Jt-o and [X,, placed at the ends B and A of the magnet; 2" A fictitious fluid trail, distributed along the line s, and having for linear density at each point as 76 BOOK SEEN. - MAGNETIC FORCES. If, in particular, the quantity pi is constant along the line s, we have [^1= !^o= |J-, X == o; formula (i) will become The magnet will exert the same external actions as two equal and opposite sign quantities of fictitious fluid placed at its two ends. Such a magnet is called a magnetic solenoid. We have seen (Chap. I) that extremely long steel needles behave much like magnetic solenoids. Coulomb and the experimenters who, after him, dealt with the study of linear magnets that can be made with steel needles, implicitly admitted in their research that the quantity [j. was equal to o at both ends of a linear magnet. This hypothesis, expressed by the equalities 1^0= o, r^i = o, seems to contradict this fact, that very long steel needles behave much like magnetic solenoids. But we will see later that this contradiction is only apparent. We can therefore adopt the following hypothesis: in realizable linear magnets, the quantity p. is equal to o at the extremities, even if we have to verify the consequences of this hypothesis experimentally. Under this assumption, equalities (i) and (2) give ds. We can see that the study of the properties of the magnet under consideration comes down to the determination of the value of the quantity X at each point. We will examine the principles behind this determination. LINEAR MAGNETS. Let BA {fi g. 1 1) a line on which the fictitious fluid is distributed with a continuous linear density. Let P be a point of this line and \ the value of the linear density at this point. Fis. II. 1° Suppose that the point P is not one of the ends of the line BA. Through this point, let us lead a normal PN to the line BA. On the normal PN, let us take a point II whose distance o to point P is infinitely small. Let us take the line BA as director of a channel surface whose radius is 8. Let us then take on the line BA, on both sides of the point P, two points M, M', whose distance ds is infinitely small. Through the points M, M', let us lead two straight sections of the channel surface. Let us apply Gauss' lemmas to the small cylinder bounded by these two sections. The two bases, thanks to the smallness of o, provide the sum of the normal components of higher order infinitesimals. On the lateral surface, the normal component of the magnetic action can be considered as having the same value Fj at any point. The sum of the normal components has therefore for value ?. TTO ds Fa . These actions come from the fictitious fluid distributed on the line Bx\ with density \. We have therefore 2 710 "?5F.\ = 41^^ ds or (3) À=i8FA. 2 2" These considerations no longer apply to a point P whose 78 BOOK VII. - THE MAGNETIC FORCES. the distance to one end of the line BA is not finite. Indeed, in the vicinity of this point, Fj, could no longer be a continuous function of. Let us therefore examine directly the particular case where the point P would coincide with one of the ends of the magnet AB, the end B for example (Jig- 12). Fi£ Through the end B of the magnet BA, let us lead a plane P normal to the magnet. Let BA' be the symmetrical line of the line BA with respect to this plane P. On the line BA' let us distribute some fictitious fluid, so that in two points M, M', symmetrical with respect to the plane P, the linear density "k of this fictitious fluid has the same value. Let us take, in the plane P, a point II situated at a distance 8 from point B. Let Fj^ be the component along BII of the action exerted at point n by the fluid spread along line BA. The action exerted at the same point by the fluid spread along line A'BA will have as its component, along BIT, cp,j^2F^^-. Now, according to the above, we have -, I ^ A = - 0 "N , 2 . OR (4) X = 3Fn. Equalities (3) and (4) lead to the following proposition: A line ^K carries fictitious fluid whose linear density is \ at point P. A pole II, equal to V unit, is located at an infinitely small distance 8 from point P on a normal N to the. CHAP. VI. - LINEAR MAGNETS. 79 line BA led through this point. We measure the component Fjj along the normal N of V action exerted at point II. If the point P is not an end of the line BA, we have - 8Fn, 2 and if the point P is an end of the line BA, we have The product ôFj cannot be determined experimentally at an infinitesimally small distance 8 from the line s] the distance 8 at which this product is measured is very small; however, it must still be large enough for the transverse dimensions of the magnet to be neglected. We are thus led to modify the previous statement, and to replace it by the following one: The value of V^ is determined at a very small distance from Vaimant, though large compared to the transverse dimensions of Vaimant. If S is this distance, we have X = -8Fn, k being a factor equal to unity for points whose distance to the end of the magnet is large compared to S, variable between i and i for points whose distance to the end of the magnet is of the order of o and substantially equal to i at V end of the magnet. Let us imagine that we have represented by the curve a^y i^fig' ^3) the value of the product - oF" for each value of 5. X will be represented by the curve a' [S'y', coinciding substantially with the previous one for values of s that differ substantially from o or L, while at the extremities this new curve will have double the ordinates of the previous curve. These various principles had been very well perceived by Coulomb (' ), whereas they were often ignored by the physicists who followed him, notably by Gaugain (2). (' ) Coulomb, Septième Mémoire sur l'électricité et le magnétisme: Du magnétisme, art. XIX {Mémoires de l'Académie des Sciences pour 1789, p. 473) (") Gaugain, Mémoire sur la distribution du magnétisme dans les électroaimants {Annales de Chimie et de Physique, b' série, t. XI, p. 5; 1877). 8o BOOK VU. - THE MAGNETIC FORCES. The determination of the linear density ). of the dummy distribution is reduced, from the above, to the determination of F^. . Fis .3. To determine Fj, Coulomb used the methods indicated in the previous chapter for any magnet. In the case where the linear magnet is rectilinear, there is a great advantage in substituting for Coulomb's methods an elegant method, based on the properties of electromagnetic induction. This method was devised by Van Rees ('); it was later used by Gaugain (-). A small metal circle, of radius S, can be moved so that its axis constantly coincides with the axis BA of the magnet. This circle is connected by a double metal wire to a very distant ballistic galvanometer {fig. i4) This circle is first at rest, its center being at O. It is moved rapidly, so as to bring its center to O', and it is stopped again. An induction current of short duration runs through this circuit. The circuit being in the neutral state at the beginning of the modification as well as at the end, this current sets in motion, at each point of the circuit, the same quantity of electricity Q, a quantity that will be measured by the impulse given to the ballistic galvanometer (see Book XIII, Chap. VII, § 2). To calculate this quantity, we can reason as if the (' ) Van Rees, Ueber die Verthellung des Magnetismus in Magneten {Poggendorff's Annalen, t. LXXIV, p. 2i3; 1849). (" ) Gaugain, loc. cit. CHAP. VI. - LINEAR MAGNETS. The current that flowed through the circuit had been uniform at all times. The current having an intensity equal to o at the beginning and at the end of the modification, there is no need to take into account the induction of the circuit on itself. The magnet is not influenced by any current at the beginning and at the end of the modification; Fig. i4. its magnetization is therefore the same in both cases, and we can, in order to calculate the quantity Q, suppose that this magnetization has remained invariable during the whole duration of the modification. We are therefore led to study an induction phenomenon produced by a magnet that is invariant in shape, position and magnetization in a circuit that moves while remaining traversed by a uniform current. Now, as we will see later (Book XV, Chap- III), to determine, in this case, the product RQ of the quantity of electricity set in motion by the resistance of the circuit, we imagine that each magnetic mass M of V magnet exerts, on each element dl of the conductor, a force having the magnitude 4^ r^ normal to the plane of the current dl and the mass M, and directed to the right of the observer who, placed in the element dl, would look at the mass M [the direction r is counted from the element dl towards the mass M]; the work produced by this force in the displacement of the conductor is calculated; it is equal to - RQ. Let us apply these results to the particular case we are dealing with. Let M{Jig. i5) be one of the fictitious magnetic masses distributed on the line BA and dl = mni an element of the circuit. The angle (r, dl) D. - II. 6 8-2 BOOK VII. - THE MAGNETIC FORCES. is straight, and the force F has the value & M 471 r^ dl. This force is normal to the plane M mm'. The radius Om or S of the circle being very small, the direction Mm differs very little from the direction MO and the plane MmO is almost normal to the plane M mm'. Fig. i5. ..Ill^ The force in question is therefore located approximately in the plane MmO; it is approximately perpendicular to Mm. Let & be the angle mMO, the force in question will have, on the dii^ection BA, a projection substantially equal to ^ Msinô 471 ^2 dl. The set of forces acting on the small circle has, according to BA, a resultant which has the value ^S -^ M S Msin9 the sign \^ extending to all the masses M which compose the magnet. Now it is easy to see that we have The force in question can therefore be written " m. When we bring the center of the small circle from O to O', it ef CHAP, VI. - LINEAR MAGNETS. 83 fects a work So we have fE = ^8 f Fn ds. 2 Jo This formula will easily allow to determine, for each value of s, the value of oFj, and, consequently, the value of 1. The experimentally found results can be represented by the formula (5) l^a(ks-k^^-^), a and A" being two constants which depend on the nature of the steel used, the means used to magnetize the needle, and the diameter of this needle. Equalities . ten (6) j,^_^(A-.+ A-L-_A-i._,) then give logA: From this formula we deduce, as we should have expected, 1^1= o. This formula is due to Biot (*). Green (-) tried to justify it theoretically. When the needle is very long and very thin, ). has sensitive values only for values of s or of (L - s) which are small in relation to L; the needle then behaves, for slightly distant points, like two equal magnetic masses of opposite signs placed at its two ends. This proposition, in conformity with experience, is not, as we can see, in contradiction with the hypothesis expressed by the equalities (7) (^0=0, îXi = o. (') Biot, Traité de Physique, t. III, p. 76; Paris, 181G. {") Green, Essay on the application 0/ mathcmatical Analysis to the theories of electricity and magnetisni (Nottingham, 1828). 84 BOOK VII. - THE MAGNETIC FORCES. This hypothesis, expressed by the equalities (7), serves as the basis for all the experimental studies that have been used to establish formula (6). It would obviously be interesting to be able to submit formula (6), or one of its consequences, to an experimental verification which would not make use of the hypotheses (7). The comparison of magnetic moments provides this verification. According to formula (6), the magnetic moment of a magnetized needle has the value (8) D\l = i'''*=n^- [<-"''-'>' This formula must be able to represent the magnetic moment of needles of different lengths, having the same section, made of the same steel and magnetized in the same way. M. Bouty (' ) has submitted this formula to the control of the experiment by means of an ingenious method, which allows to find very quickly the ratio of the magnetic moments of two magnetized needles. The two magnets BA, B'A' are suspended from the same equi])age [Jig- 16) so that they are both horizontal. They make Fis. 16. B-^^^^ B^--""''^ --- "T ^^<^ h h between them an angle 0. When these two magnets are in equilibrium, the direction BA makes an angle a with the direction av of (') Bouty, Études sur le magnétisme, i" Partie {Thèse de Doctorat, Paris, 1874; Annales scientifiques de l'Ecole Normale supérieure, 2' série, t. IV, 1873). CHAP. VI. - LINEAR MAGNETS. 85 the magnetic meridian. If OTL and DÏL' are the magnetic moments of the two magnets, the equilibrium condition is DlLsinar^ OlVsin(0 - a). In particular, if the two needles make a right angle between them, this equality will become tangqt = 011' D\l The determination of the angle made known the ratio of the magnetic moments of the two magnets. The results of Mr. Boutj's experiments agree approximately with the numbers provided by the formula (8), 86 BOOK VII. - THE MAGNETIC FORCES. CHAPTER YII. SOLENOIDAL AND LAMELLAR DISTRIBUTIONS. § 1 - On the solenoidal magnetic distribution. SirW. Thomson (^) has given on the distribution of magnetism inside magnets some theorems which, without having the practical importance of the propositions relating to the superficial distribution, are however remarkable by their elegance and their generality. M. Betti (^) gave a very nice exposition of the theorems of Sir W. Thomson. The present Chapter will be devoted to reproduce this exposition. We will take as a starting point the following theorem, which is due to Jacobi: If three functions X, alb, 3 are, in a certain space, uniform, finite and continuous as well as their partial derivatives of the first order; if, moreover, they verify, in any point of this space, V equation dX dlftj d^ ' dx dy dz there are two functions ui, v, which are, in this space, uniform, finite and continuous, as well as their partial derivatives (' ) Sir W. Thomson, A mathematical theory of magnetism {Proceedings of tlie Royal Society, June 1849. ~ Sir W. Thomson's Reprint of papers on electrostatics and magnetism, Art. XXIV, Chap. V). {') Betti, Teorica délie forze newtoniane e sui applicazioni aW elettrostatica e al m,agnetismo, Chap. III, § 3; Pisa, 1879. CHAP. VII. - SOLENOIDAL AND LAMELLAR DISTRIBUTIONS. of the first order, and which are such that Von has 87 {■^) dfx dv "ilû - Ji Jl J -, '. dy dz dy dz dz dx dz dx ç^ _ àix dy diJ. dx dy dx dy Having admitted this theorem, let us consider a magnet and suppose that the three components X, alî), G of the magnetization verify, at any point of the magnet, the condition (i). Let m, n be two variable parameters and consider the two families of surfaces (3) (4) V = n. Two of these surfaces, one of the first family^ the other of the second, intersect along a line; through each point M(^, y, z) of the body, there passes one of these lines and only one, since this point corresponds to a single value of [i. and to a single value of v. This line moves and deforms in a continuous way when the two parameters m and n change value in a continuous way. Let N and N' be the normals led by the point M to the surface H = m is 1 a on face which pass through this point; let a, p, y be the cosine directors of the normal N and a', P', f' the cosine directors of the normal N'. We will have (5) ^\2 ày, [( dx m djx BOOK VII. - THE MAGNETIC FORCES. r/d^/V /à^Y /(?v\2"12 d^i L(^; ^w) "^U; J " = ^' [(1^) a2 + p+Y' = i> a'2+p'2+Y'2=i transform this equality into (8) 01I2=n(JL.nv.[i-(aa'-4-pp'+YT')'] CIIAP. VII. - SOLENOID VLES AND LAMELLAE DISTRIBUTIONS. 89 Consider the four surfaces ^ = m -h A/n, V - re -t- \n. They cut, in the body, an infinitely loose channel, which can be compared to a linear magnet. According to the previous theorem, along this linear magnet, the magnetization is longitudinal. Let M(x, y, z) be a point on the line j fx = m, I V = n. Through point M, let us lead a plane normal to this line. This plane meets respectively the three lines { ix = m -h Im, \ [X = 7)1, i [x = m -+- Am, ( V r= /i ; ( V = n -+- A/t ; ( v = n 4- A/i to the points P{.T+\x, y-h^y, -4- A3), V{x -\- ^"x, y ■+■ \"y, z h- A"^). The parallelogram MPP'P" is the right section, at point M, of the infinitely untied channel. If we designate by Q the area of this parallelogram, we will have (9) We have obviously Û2= MP .MP' .sin^PMP'. -^ Aa7 -+- -r^ A V + -i- A^ = Am, dx dy -^ dz -- \x -- \y -^ ~ \z dx ôz Moreover, the plane normal to the point M at the intersection of the two surfaces a = m, / = /i is the NMN' plane, whose equation is (PY'-Yp')(X-a;)4-(Ya'-aY')(Y-^)-f-(a?'-pa')(Z-3) = o. GO BOOK VII. - THE MAGNETIC FORCES. The point P being in this plane, we have (Py' - YP')Aa7 -I- (ya'- ay' ) A^ + ( a^' - ^a')\z = o. By means of equalities (5) and (6), we see that the three equalities verified by A.r, Ay, A2 are the following: a A27 -h p Aj' 4- Y A^ = -^ A/71, (n{x)2 a.' Acp -\- '^' \y -h ^' ^z = o, (Py'- yP')A^ +(Ya'- aY')Aj +(a!3'- ^oc')\z = o, which gives (n(x)H(PY'-TP')^ + (ï"'-aï')^-t-(^?'-pa')2] ^, _ [a'(Y"'-"r)- P'(Py -#)]Am (npi)^[(pY'-Y3')'-*-(Y=''-='Y')' + (a?'-P"'/'] and, therefore, MP ^ = ( Air )2 -(- ( A^)2 -+- ( Az )2 I nia [(Py'-yP')^ + (y"'-"y')' + (^?'-P"')'? x![P'(ap'-Pa')-Y'(Y"'-"Y')? + [y'(Py'-yP')- "'("?'-?='')]' + [a'(Ya'-aY')-P'(PY'-Y?')]'l(A'")^ Let L be the line ^ = m, V = n. The normal to the plane LMN' being designated by (Dî>), we have cos(^, a7)=p'(aP'-?='')-Y'(Y"'-"Y'). cos(^, jK)= y'(Py' - Y?')- <^'("^'- P"'), cos(3î,, z ) = a'(Ya' - "y')- P'(Py'- Y?') Therefore, according to the equality cos^{SiZ>, x)-^- cos^ {Sf^, y) -h cos^{2(Z>, z)-i, CHAP. VII. - SOLENOIDAL AND LAJIELLAR DISTRIBUTIONS. equality (lo) will become Mr=J- ^'"'^' or (it) MP = =- ^ ^ Dfji [i - ("a'+pp'+YY'j^]" We have, similarly, (12) MF' = nv [i_(aa'+ pP'-t-Yï')^] Finally (i3) sin2PMP'=i - cos2NMN'=i- (aa'+ P^'+yy')^ By virtue of equalities (i i), (12) and (i3), equality (9) becomes o,^ (Am.An)^ n|x.nv.[i-(aa'- PÎ3'+ yy')'] ' equality which, compared to equality (8), gives (0ÎLÛ)2 = (Am.An)2. The product DÏLQ thus keeps the same absolute value all along the infinitely unbound channel considered; this channel constitutes what we called, in the previous chapter, a magnetic solenoid. We therefore arrive at the following proposition: If the components of the magnetization satisfy at all points of a body the condition dX dlJi) d^ there are two families of surfaces cutting the body into infinitely loose channels, each of which is a magnetic solenoid. Sir W. Thomson gave the name of solenoidal magnetic distribution to a distribution whose components verify equality (i) at any point. We have seen (Chap. IV, § 2) that such a distribution has remarkable properties. dlft, from dz at = from dX dx dz = o, dX d\S\, 'ày~ dx = 0. O'i BOOK VII. - THE MAGNETIC FORCES. § 2 - On the simple lamellar distribution. Let us now assume that at any point of a simply connected magnet the components JU, i)î), G of the magnetization are finite and continuous uniform functions of x^ y, z, and that the same is true of their first order partial derivatives. Let us further assume that these derivatives verify the conditions ('4) In this case, there will exist a uniform, finite and continuous function z>(x,y,z) such that we have (.5) X=p-, ^.= J, 3=^?. dx dy dz Let us consider a variable parameter/'. The surface (.6) '^=f is either a closed surface contained entirely within the magnet, or a limited area having its contour on the surface of the magnet. Through any point of the magnet, one and only one of these surfaces passes. When the parameter/" varies in a continuous way, this surface deforms in a continuous way, so that it sweeps the whole space occupied by the magnet. Let N be the normal at a point on the surface S represented by equality (i6), this normal being directed on the side where /va is increasing. Let a, ^, y be the director cosines of this normal. We will have (.6) (ncp)'a = g, (ncp)'p = g, (ncp)2.^ = g. According to these equations, the quantities ".l., -\S\>, e are, at each point, proportional to the quantities a, [3, y. Hence this theorem: U magnetization is, at each point of the body, normal to the surface S which passes through this point. CHAP. VIT. - SOLENOID AND LAMELLAAR DISTRIBUTIONS. gS The equalities (i5) give (17) oiL2=no. Let us lead the surface S represented by the equation then the surface S' represented by the equation Let M be a point on the surface S. Through this point, let us lead a normal to the surface S and limit it to the point M' where it meets the surface S'. We will have or, according to the equalities (16), (ncp)2.MM'' = A/. This equality, compared to equality (17), gives D1(L.MM'= A/.' The product of V magnetization at a point on the surface S by , the normal distance between the two surfaces S and S' has the same value at any point on the surface S. We will call a magnetic sheet a magnetized layer, included between two infinitely close surfaces, whose magnetization is, at each point, normal to the layer and in inverse proportion to the thickness of the layer. We see that the two surfaces S and S' limit, in the magnet, a magnetic sheet, open or closed, which leads us to the following theorem: If the three components of V magnetization verify, at any point, the conditions o, ('4) dl)î, dZ 'dz dy dZ d\^ dx dz d^o ai)b dy dx there is a family of surfaces decomposing V magnet into 94 BOOK VII. - THE MAGNETIC FORCES. infinitely thin lamellae, closed or open, each of which is a magnetic sheet. Sir W. Thomson gives the name of simple lamellar distribution to a similar magnetic distribution. For conditions (i) and (j4) to be verified at the same time, it is necessary and sufficient that there exists a function cp verifying both conditions (i5) and the condition Ao = o; therefore, for a distribution to be both solenoidal and simple lamellar, it is necessary and sufficient that the three components of V magnetization are equal at each point to the first order partial derivatives of the same harmonic function of the coordinates. We will see in the next book that, according to Poisson's theory, a piece of soft iron, magnetized by influence, would be found in these conditions. § 3 - Distribution on any magnet. Let us now consider a magnet inside which the three quantities o,l>, i)l>, S are subject only to being uniform, finite and continuous, as well as their first and second order partial derivatives. The identity d fdA, c)l)\ = o, together with Jacobi's theorem, which we stated at the beginning of§l, shows us that there always exist two uniform, finite and continuous functions a (a:?, j', s), v(^,y, :;), as well as their first order partial derivatives, such that one can write (i8) d /()ilî, d^\࣠dz\ to/from ày ) '^ ày \ dx Mo\ d /dJ{s> dz )~^ dz[ dy c)\)l dx dx\\\, dz dB dy ten dv ^ dfdl' ten dv dz dy d3 dx d.X. dz ten d^ ~ dz tox ten c^/ dx dz dX dx ten d i dx dy ten d'/ dy dx CHAP. VU. - SOLENOIDAL AND LAMELLAR DISTRIBUTIONS. <)5 These equalities (18) can still be written dz\'' '' djy) (>r \' àz ) dx \^ dz j dz \ ^ dx ^ fx_,^J^\ = l(,, J^ dx oivu do dfx dx dx Kl, do dwhere m is a variable parameter, which passes through this point. The magnetization intensity is determined by the equality (>n 0112 = vî.n-ji. 96 BOOK VII. - THE MAGNETIC FORCES. Consider the surface S, represented by the equation JL = /n, and the infinitely close surface S', represented by the equation ^ = m -h A/?i. These two surfaces limit a lamella, whose thickness MM' a for value -.■■■ A/n (22) MM'= ■ -. Comparing equalities (21) and (2a) gives 0\i.MM'= \l'.^m\. The product of the magnetization by the thickness of the layer does not have the same value at every point of the layer; its value is proportional, at each point, to the value of the function v at that point. A distribution formed of similar lamellae is what Sir W. Thomson calls a compound lamellar distribution. Thus, any continuous magnetic distribution can be viewed as the superposition of a simple lamellar distribution and a compound lamellar distribution. If the components of the magnetization verify the conditions (i4)j the simple lamellar distribution remains alone. .5 4 - Decomposition of any distribution into a simple lamellar distribution and a solenoidal distribution. Let (x'jy'j z') be any point of the volume occupied by the magnet; X', i)b', S' the components of the magnetization at this point; /- the distance from the point {x^y, z) of space to the point {x' , y' , z'). Consider three functions of x^ y, z defined as follows (23) 1 i.=J -ch, integrations extending to the entire volume of the magnet. CIIAP. VII. - SOLKNOIDAL AND LAMELLAR DISTRIBUTIONS. 97 Let's then pose (2|) f dG dB. a = dz -^ dH dV h - dx ~ Tz' d¥ dG c = d^ dx dF dx dG dH "^ dy ^ ~dz ' . the identities from Ty db ■ dz dx da àz"^ from dx dh tox da dz (■>.'>) But, according to the equalities (aS), we have, at any point of the magnet, AG ^::- 47:111), AH^ - 47r3. The equalities (aa) can thus be written d^ /de db\ ^""^---à^-idy-dz)' da de \ dz dx J dz \dx dy j These new equalities prove the following theorem: Any continuous magnetic distribution can be considered as resulting from the superposition of a solenoidal distribution and a simple lamellar distribution. 4-"-- ^-(--^' D. - IT gS BOOK VII. - THE M.VGNETIC FORCES. CHAPTER YlII. MAGNETIC MERIDIANS AND MAGNETIC PARALLELS. § 1 - Distribution of the tangential force on the surface of any body. In the study of terrestrial magnetism, the value of the magnetic declination and of the horizontal component of the magnetic intensity are determined with great care at each point of the surface of the globe; that is to say, the magnitude and direction of the tangential component of the magnetic force are studied at each point of the surface of the globe. Gauss (' ) has given, about the distribution of this tangential component on the surface of the globe, a number of beautiful theorems. Among these theorems, some are general and can be applied to the study of the tangential component of the force on the surface of any magnet. The others can only be applied to a spherical body, such as the earth. We will first study the former, then the latter. Let us therefore consider a body which we will assume to be simply connected and whose surface will not have any singularity. Letx") be the magnetic potential function of this body. This function cannot have the same value at all points on the surface of the body, because then the fictitious surface layer that is equivalent to the magnet would fulfill the conditions of electrostatic equilibrium. As this layer contains as much positive fluid as negative fluid, it would have a zero density at all (') G. -F. Gauss, Allgemeine Théorie des Erdmagnetismus {Besultate aus der Beobachtungen des magnetischeti Vereins im Jahre i838, published by Gauss and Weber; Leipzig, iSSg. - Gauss, Werke, Bd, V, p. 121). CHAP. VIII. - MAGNETIC MERIDIANS AND MAGNETIC PARALLELS. 99 pointl of the body surface, and the magnet would have no action at the points outside it. We exclude this particular case from our research. The magnetic potential function thus varies at the surface of the magnet and its variations necessarily admit an upper limit A and a lower limit B. As the magnetic potential function is a continuous function, each of the values between A and B, without omitting either the value A or the value B, is taken by it at the surface of the magnet. Let C be a value between A and B. The points on the surface of the magnet where the potential function has values between A and C must be separated from the points where the potential function has a value between C and B by a region where the potential function has the value C. It will always be possible, through the points on the surface of the magnet where the potential function has the value C only, to pass at least one closed line. It may be that these points do not only form this closed line, but that some of them are isolated, or form line segments, or cover spaces of finite extent. Equality (i) P = G therefore defines, in general, a closed line drawn on the surface of the magnet; this line is called a magnetic parallel. In space, equality \'^ = G defines a level surface to which the magnetic force is normal at each point. From this point on, it is easy to see that the tangential component of the force is, at each point of the body, either zero, or normal to the magnetic parallel which passes through this point. We will give the name of magnetic meridian to any line drawn on the surface of the magnet and tangent at each of its points to the tangential component of the force. The previous theorem can then be stated as follows: The magnetic meridians and the magnetic parallels form, on the surface of the magnet, two families of lines cutting this surface into infinitely small rectangles. 100 BOOK Vir. - THE MAGNETIC FORCES. Let's give successively to the parameter C the values C, G -I- £, G-i-2£, ..., G-+-{n - i)z, G-t-/i£, ..., £ being an infinitely small quantity. By means of equation (i), we draw magnetic parallels on the surface of the magnet Po) f*l) P2) ---) P" - lî P/ii --- infinitely close to each other. Let M be a point on the parallel Pn-i - Let H be the tangential component of the magnetic action, this action being counted positively from parallel P"_i to parallel P". The force H is tangent to the magnetic meridian of the point M. This magnetic meridian meets at M' the parallel P". We have then, according to the fundamental properties of the magnetic potential function, -CPCM)- '(?(M') £ H = MM' MM' Moreover, if we limit ourselves to considering the infinitesimals of the first order, MM' is, in general, equal to the distain B of the two infinitely close parallels, P"..4, P". Thus we have (.) H = -^ The tangential component of the magnetic action is, at each point, measured by the inverse of the distance between two magnetic parallels; it is, at each point, normal to the magnetic parallel and directed in the direction in which the magnetic potential function decreases. Along the same magnetic meridian, traversed in a well-defined direction, the tangential component of the magnetic action, counted positively in this direction, does not necessarily have a constant sign; but, because of its continuity, it must keep the same sign all along the continuous arcs separated from each other by points where it has the value o. It is such a continuous arc, along which the tangential component of the magnetic action keeps an invariable sign, that we will name from now on a magnetic meridian. CHAP. VIII. - MAGNETIC MERIDIANS AND MAGNETIC PARALLELS. 10 1 A magnetic meridian cannot form a closed line. Let us denote by f/5 the arc element of a magnetic meridian; according to the previous definition, the quantity -r- cannot change sign along the same meridian. The integral / extended to a portion of a meridian or to the whole meridian, can never be equal to o. Now, the function "Ç being a uniform and continuous function, the preceding integral, extended to any closed curve, is equal to o. A meridian cannot therefore be closed by itself. The same meridian cannot intersect the same parallel more than once. Indeed, since the function V has the same value at all points of the same parallel, if a meridian intersects a parallel more than once, the integral / - ds. Os ' extended to a segment of the meridian between two encounters with this parallel, would be equal to o, contrary to what we have just seen. Thus, a magnetic meridian necessarily forms, on the surface of the body, a segment limited by two points; the potential function does not have the same value at these two points - at these two points, the tangential component of the magnetic action is zero, and, consequently, the magnetic action is either zero, or normal to the surface of the body We will call the magnetic pole a point on the surface of the body where the tangential force is equal to zero. This word takes here a meaning in conformity with the one attributed to it in the study of terrestrial magnetism, but quite different from the one attributed to it, in general, in the part of Physics which deals with magnets. The previous theorem shows us that any magnetic meridian starts from a pole and ends at a pole. Let P be a magnetic pole; let s be an arc drawn on the surface and I02 BOOK VII. - THE MAGNETIC FORCES. passing through the pole P. The component of the magnetic action along the tangent to the arc s has the value ~~ds' For the point P to be a magnetic pole, it is necessary and sufficient that this component is equal to o whatever the arc s. Therefore, in order for a point to be a magnetic pole on the surface of the body, it is necessary and sufficient that the derivative of the potential function along the arc of any curve traced on the surface and passing through this point is equal to o at this point. Therefore, three different circumstances can arise in a magnetic pole: 1° It can happen that in a magnetic pole the magnetic potential function has a smaller value than in the neighboring points of the body surface. In this case, if we use equality (2) to determine the tangential component of the magnetic action at a point near the pole, we find that this component is directed toward the pole. The pole attracts a mass of southern fluid placed in its vicinity and subject to move on the surface of the body. The pole is a north pole. 2° It can happen that in a magnetic pole the magnetic potential function has a higher value than in the neighboring points of the body surface. In this case, if we use equality (2) to determine the tangential component of the magnetic action at a point near the pole, we find that this component tends to move away from the pole at its point of application. The pole pushes away a mass of southern fluid placed in its vicinity and subject to move on the surface of the body. The pole is a south pole. 3° It can happen that in a magnetic pole lines pass which divide the domain of this pole in several regions, the number of which is at least four; some, R, R', ... . . where the potential function has a higher value than at the pole; the others, p, p', . - . . where the potential function has a smaller value than at the pole. A mass of austral fluid, placed in the domain of the pole and subjected to move at the surface of the body, is repelled by CIIAP. VIII. - MAGNETIC MERIDIANS AND MAGNETIC PARALLELS. Io3 the pole if it is located in one of the regions R, R', . and attracted speaks pole if it is in one of the regions p, p', Le The pole behaves like a south pole for the first regions, like a north pole for the others. We will say that it constitutes a mixed pole. We know that on the surface of any body there is at least one point where the potential function reaches its maximum value A and one point where the potential function reaches its minimum value B. Therefore, on the surface of any magnetized body, there is at least one north magnetic pole and one south magnetic pole. If, on the surface of a magnetized body, there are two poles of the same name, there is at least one mixed pole on the surface of this body. Let us imagine, for example, that on the surface of a magnetized body there are two south poles, P and P'. These two poles correspond to two values A and A' of the potential function which are two maxima. Let us suppose, to fix the ideas, A l A'. Let us consider the locus of the points for which we have i:^ = A'- £. We can always take the quantity s small enough so that (A' - s) is greater than the lower limit B of the values of the potential function on the surface. There will then be, on the surface, points corresponding to this value of the potential function. Among these points are all the points of a certain closed curve separating the regions R in which t? has a value higher than (A' - s), from the regions R' where t? has a value lower than (A' - e). The points P and P' are in the region R. We can always take t small enough so that the domain of point P' inside which 'Ç lies between A' and (A' - e) is smaller than any given surface; small enough, therefore, that this domain does not contain point P. For such a value of e, the region R must decompose into at least two linearly related parts: one p, surrounding the point P', the other Jl, containing the point P within it. Let us increase e, so that (A' - e) tends to B. The I04 BOOK VIT. - THE MAGNETIC FORCES. regions p and ^ will tend to cover the whole surface of the body. The contours of these two regions will therefore not always remain at a finite distance. It may happen that these two contours remain distinct until (A' - e) reaches the value B, and then they merge in all their extent. In this case, there would be on the surface of the body a whole closed line where \'^ would have the value B, i.e. a whole line of north poles. Let us exclude this exceptional case. The two contours will, in general, take common points when (A' - s) takes a value A" greater than B. It may be that these common points form a line of finite extent; but let us exclude this particular case. In general, for the value A" of the magnetic potential function, the two contours will touch at a certain number of isolated points. Let P" be one of these points. The line <;> = A", passing through point P", divides the domain of this point into several regions. The ones are part of the regions p and A. The potential function there has a value greater than A". For the points placed in this region, the point P" behaves like a north pole. The others are part of the region R'. The potential function there has a value lower than A". For the points placed in this region, the point P" behaves like a south pole. Point P" is therefore a mixed pole, which justifies the proposal we made. Mr. Betti (' ) has proved this beautiful theorem: If the surface of a magnetized body is a simply connected surface and if the number of magnetic poles it carries is a finite number, this number is even. Let us reproduce here the demonstration he gave: On the surface of the body there is at least one North Pole and a south pole S. Let us design, on the surface of the body, a system of C curves all starting from the south pole S, all arriving at the pole north N, none of which passes twice through the same point, C) Betti, Theoria délie forze Newtoniane e sut applicazioni ail' elettrostatica e al niagnetismo, Ch. III, § 2; Pisa, 1879. CIIAP. VIII. - MAGNETIC MERIDIANS AND MAGNETIC PARALLELS. Io5 none of which meets any other curve of the system outside the points S and N, and such that through any point on the surface of the body there passes one of these curves. This will always be possible, provided that the body occupies a simply connected space. If one travels along a curve C from point S to point N, the magnetic potential function \'> will be decreasing at the beginning of this path and still decreasing at the end. Its derivative along the arc of any curve drawn on the surface being supposed to be continuous, the sequence of values it will take along this path will have neither maximum nor minimum, or will have an equal number of maxima and minima. Let us denote by ^3,, [^2, .'. ., j3" the points on a curve C where the value of \'^ on that curve becomes minimum; by a(, a2, . . ., a" the points, equal in number on that curve, where the value of "Ç becomes maximum. The points ^,, ^05 ■■-■) '^n are assumed to be arranged in the order in which they are encountered in going from point S to point C. The same applies to the points a,, olo, ..., a^^. It is easy to see that between the points [3/_( and [3,- is always one and only one of the points a, the point a,; that between the points a, and oLi^t is always one and only one of the points |3, the point [ii,. Let's take one of the G curves for which the number n has the greatest value, and apply to it what we have just said in a general way. Let di be the locus of the points a/ and D/ the locus of the points ^i when the curve C is varied. Either the curve di never meets the two curves D/_, and D/ between which it is included; it then forms a curve closed on itself. Or it meets one of the two curves D/_,, D/ and forms a closed curve with it. Thus, there are in general, on the surface S, a certain number ni of closed d-curves; an equal number of closed D-curves and a number (n - ni) of closed , being a uniform function, takes its value after this path 5 it is the same of its derivative -r along the arc of the curve; this derivative, which is a function con I06 BOOK VII. - THE MAGNETIC FORCES. s, thus changes sign an even number of times. Thus, along any closed curve drawn on the body surface, the potential function has an equal number of maxima and minima. The function 'Ç thus presents on any of the curves d, D, dD, the same number of maxima and minima. Now, if the function 'C is minimum at a point b of a curve of, this point is a north pole. Indeed, consider all the segments of curve C that lie in the domain of pointa. On each of these segments, the minimum value of 'Ç is at the intersection of curve d, and on curve d the minimum value of t? is at point b. The value of 'Ç at point b is therefore smaller than at any point in the domain of point b. On the other hand, if the function '^ is maximal at a point a of a curve d, this point is a mixed pole. Indeed, the value of <,> at point a is obviously increasing if one moves away from point a on either of the two directions of the curve C that pass through that point, and decreasing if one moves away from point a on either of the two directions of the curve d that pass through that point. Any closed curve d therefore contains an equal number of north poles and mixed poles. Similarly any closed curve D contains an equal number of south poles and mixed poles. Now consider a closed curve dD. It contains a number of points where the value of "Ç on this curve passes through a maximum and an equal number of points where it passes through a minimum. Any point on this curve where "C^ is a minimum is a north pole if it is on the d portion and a mixed pole if it is on the d portion. Any point on this curve where "C^ is a maximum is a south pole if it is on the d portion and a mixed pole if it is on the d portion. Each of the curves d, D, dD, containing an even or zero number of poles, the number of magnetic poles on the surface of any magnetized body is even if it is finite. This is the theorem stated and proved by M. Betli. CHAP. VIII. - MAGNETIC MERIDIANS AND MAGNETIC PARALLELS. I07 § 2 - On terrestrial magnetism. To the general theorems we have just explained, Gauss added a few others which essentially assume that the magnet has a spherical shape; these theorems therefore apply to the terrestrial magnet and give us some interesting properties of the horizontal component of the magnetic action on the surface of the earth. It is these proposals that we will now demonstrate. The first of these propositions concerns the properties of the Gaussian polygon. It follows immediately, as we shall see, from the simplest properties of the magnetic potential function. Let Mo, M, be two points located on the surface of the globe. Let us suppose that a quantity of magnetic fluid equal to the unit passes from the point Mo to the point M, by any path. The work done by the actions of the earth's magnetism on this charge is independent of this path. It has for value \'^(Mo)-t;;(M,), t? (Mo), "^(M,) being the values of the magnetic potential function at points Mo, M,. Let us suppose that the path traveled by the magnetic mass is the path MoMM(, located entirely on the surface of the earth; let H be the value at point M of the horizontal component of the terrestrial magnetism; let t be the angle that the line MoMM, makes, at M, with the magnetic meridian of point M; let ds be an element of length of the line MoMM, . The work produced will have the value / M. H cos/ ds. We will thus have .M, H co%tds = t:) (Mo) - ■<) (Ml : 'M" If we take as integration contour a closed curve I08 BOOK VII. - THE MAGNETIC FORCES. traced on the surface of the earth, we will have / H cos t ds ~ o. Let us take a closed polygon on the surface of the earth, bounded by great circle arcs, and apply the previous equality to it. To simplify the calculations, let us suppose that this polygon is reduced to a triangle MoMiMo. We will then have "M, M, -,M" / ïi cos tds~^ I H cos tds-^ 1 Hcos^<^5 = o. -/>l""^Mi"^Mj Let's transform the expression i M, H cos t ds. Let us suppose that the great circle MolVl) makes an angle (o, i) with the geographic meridian, and to the east of it. Let us suppose, moreover, that at the point M of the arc Mo M) the magnetic declination, supposedly eastern, has a value o. We will then have, at point M, / = (o,i) - 0, which gives / It cos tds- I Hcos[(o, i) - 3]o?5. H and are functions of 5j but, if the two stations Mq, M) are not very far, one can approximately replace, in the previous integral, the quantity H cos[(o, i) - o] by the quantity ^ jHoCos[(o, r) - 00I-+- H,cos[(o, i) - o,]J, average of the values that the previous quantity takes at the ends of the arc MoM,. We will then have J/^ ' \T IVT ' II cos tds = - - - jllocos[(o, 1) - Oo] -h HiCOs[(o, i) - 81 1{. Mo ■^ CHAP. VIII. - MAGNETIC AND PARALLEL MAGNETIC MERIDIANS. lOCj For the triangle MqMi M2 we have (' MoMi j Ilocos[(o, i) - 8o]-^H,cos[(o, i) - Ojj (3) l -t- M1M2 I HiCOS[(l, 2) - 81] -f- H2COS[(l,2)- Sâjj ( -i- MjMo j H2COS[(2, o) - O2] -h HoCOs[(2, o) - 80] j = o. If we know the value of the magnetic declination at three points of the globe, moderately spaced from each other, and the value of the horizontal component of terrestrial magnetism at two of these points, we can, by this formula, calculate the value of this horizontal component at the third point. Gauss applied this formula to the triangle formed by Gœltingue, Milan and Paris. Taking as unknown the value of H in Paris, he found H = 0,51696, while direct observation gave 11 = 0,51804. Such a precise agreement demonstrates the usefulness of the formula we have just established. It will allow to determine the value of H at a station where an observer will have determined only the declination. Let M be a point located on the surface of the earth, or outside the terrestrial sphere. Let us denote by r the distance of this point from the center of the earth; by \ its colatitude, counted positively in the northern hemisphere; by J^ its western longitude. The three coordinates r, .^, \ fix its position. The magnetic potential function at this point is a function t? (/-^ 4^^ X) of these three variables. The magnetic action at point M can be broken down into three rectangular directions: the first component Z is assumed to be directed towards the center of the earth; the second Y is horizontal, located in the meridian of point M and directed towards the north. The third X is perpendicular to the two previous ones and directed towards the west. I d-ç X = r sinX ^' Y^ Z ^ ùr' 110 BOOK VII. - THE MAGNETIC FORCES. We will then have (4) These three equalities apply, in particular, to the case where the point M is located on the surface of the earth. In this case, X and Y will be the components of the projection H on the horizontal plane of the terrestrial magnetic action, so that we will have (4') H==(XM-Y2)^. The declination, assumed horizontal, will be given by the formula (i ) tango = ^. Finally, the inclination is given by the formula (D tang.--^. We can therefore, if we know the expression of tp as a function of /-, ). and C-, calculate the elements of terrestrial magnetism at any point. We have seen (Chap. V, § 2) how one can determine '^. We will first indicate some interesting consequences of the formulas we have just established: i" Let us suppose that we know the expression of Y at all points on the earth's surface, or, in other words, that we know how to express Y in terms of J^ and X. Let us take, on a certain meridian, a point M corresponding to a certain value of X; let us form the sum of the quantities Y cfk for all the points included between the north pole and the point M, and let us designate this sum by T : T will be, like Y, a known function of 4^ and X. CHAP. VIII. - MIGXETIC MERIDIANS KT P.VKVLLELES MAGNETIQUES. I I I We will have atT I dp K ^X â(™ -W-^ or, by referring to one of the equalities (4), We deduce This takes place at any point on the meridian which has longitude 4^. Let us make this point tend to the geographic north pole; T will tend to o, P to the value P^ of the potential at the north pole; 'ï-(J^) will not vary. It follows that 'f{X^) is a quantity independent of 4^, constant and equal to - "C^oThe previous formula then gives us On the other hand, T_ dp _ _j_ ÔT R sinX ôJt^ sinX âi^ Let's refer to the definition of T, and we will find V ' r^'^^ ^ Hence this remarkable theorem of Gauss: If Voii knew the horizontal northward component of the earth's magnetism at all points on the globe, we also know the horizontal westward component, and therefore the total horizontal component. Let us suppose that X is known at all points of the globe; X will then be a known function of X and "1. Let us take on a parallel two points Mo, M corresponding to values 4j) and -Ci of 4^ and let us pose U=- f Xsinldil; ■\0 112 BOOK VII. - MA.GNETIC FORCES. U will be, like X, a known function of (^and \). We will have dV . , -- = A sinÀ OR, according to one of the equalities (4), d\] __} à-Ç From this, we deduce or On the other hand, A(Ru_i_x;;)=.o Y- ' - " R d\ ' The previous formula, dilTerentiated with respect to A, can therefore be written or, by referring to the expression of U and posing The knowledge of X is therefore not sufficient to determine Y, since the expression of Y still contains an indeterminate function of the colatitude. But, if we know the value of X at all points on the globe, and the value of Y at all points on a curve passing through the two geographical poles, the function <}(X) will be known, and formula (6) will determine the value of Y at all points on the globe. Hence, I have the following theorem: If the value of the horizontal component of the magnetism directed towards the west is known at all points of the globe CHAP. VIII. - MAGNETIC MERIDIANS AND MAGNETIC PARALLELS. It3 and if Von knows, in addition, at all points of a line passing through the two geographical poles the value of the horizontal component directed towards the north, one can determine at all points the value of this last component. This theorem is still due to Gauss. I). - It BOOK VIII. MAGNETIZATION BY INFLUENCE ACCORDING TO THE FISH METHOD. CHAPTER ONE. CONDITIONS OF MAGNETIC EQUILIBRIUM. g 1. - Basic laws of influence magnetization. There are bodies whose magnetization varies so slowly when the conditions in which they are found are modified, that Ton is led to imagine bodies whose magnetism would be absolutely invariable. Such bodies, which real magnets approach more or less without being absolutely identical, are what we will call permanent magnets. On the contrary, we will call perfectly soft bodies those whose magnetism varies with the circumstances. The problem of magnetization by influence can then be stated as follows: A perfectly soft body being placed in the presence of given permanent magnets, what will be the permanent distribution of magnetism on this perfectly soft body? This permanent distribution is called an equilibrium distribution. Poisson was the first to try to equate the problem Mb BOOK VIII. - THE THEORY OF FISH. of the magnetization by influence ('); it was based on the following assumptions: Magnetic phenomena must be explained by the properties of two fluids, one positive or southern, the other negative or northern, which act on each other in the same way as electrical fluids. A perfectly soft body consists of extremely small magnetic particles, separated from each other by non-magnetic gaps. Each magnetic particle contains equal amounts of both fluids, whose distribution on the particle is freely variable. The magnetic fluids cannot move through the non-magnetic medium that separates the particles. On each particle, magnetic equilibrium is established when a mass of magnetic fluid equal to the unit, placed by thought at any point inside this particle, does not undergo any action from the magnetic fluids distributed either on this particle or on the rest of the system. For isotropic bodies, the magnetic particles are spherical. From these assumptions, by reasoning that contains several serious inaccuracies (-), Poisson has managed to deduce the following consequences: Let (^, y, z) a point of the perfectly soft isotropic body; X, i)î>, 3 the components of the magnetization at this point; 'Ç the magnetic potential function at this point of all the magnetism spread on the system. We have (i) x= - k^, 1)1,= - A-, e=-k-/-, ^ ' ax oy oz (' ) Poisson, Memoir on the theory of magnetism. Read at the Académie des Sciences on February 2, 1824. - Mémoires de l'Académie des Sciences, years 1821 and 1822, t. V, p. 247-338. (^) For a critique of Poisson's theory, we refer to Étude historique sur la théorie de l'aimantation par influence (Annales de la Faculté des Sciences de Toulouse, t. II; 1888). CHAP. I. - CONDITIONS OF EQUILIBRIUM. II7 k is a coefficient named magnetization coefficient, which depends exclusively on the nature of the body perfectly soft at the point (^,jK, z). According to the theory of Vo'i&son^ the coefficient k could not be negative. Moreover, it could not exceed a certain limit. These two conditions, imposed on the coefficient /c, run into experimental difficulties. The value found for k by studying the magnetization of soft iron exceeds the limit imposed on this coefficient in Poisson's theory. The properties of diamagnetic bodies, discovered by Faraday while studying bismuth, seem to require that the coefficient A" have, for these bodies, a negative value. The difficulties, either analytical or experimental, encountered by Poisson's theory led Sir W. Thomson (' ) to reject the whole edifice of hypotheses and reasonings taken by Poisson as a basis for the equalities (i), and to admit purely and simply these equations at a liter of hypotheses. Sir W. Thomson does not impose any restriction on the coefficient k. It can be positive, and the soft body is then said to be magnetic or paramagnetic, depending on whether the coefficient k is large or small. It can be negative, and the body is then said to be diamagnetic. We shall return, in Book IX, to the principles that can be used to establish the laws of magnetization by influence. For the moment, without discussing the accuracy of equations (i), we shall deduce a series of consequences. We shall see later how useful the results obtained will be to us. § 2 - Equation of the problem of magnetization by influence. Poisson showed how, from equations (i), one could deduce the equation of the problem of magnetization by influence, in the case where, the perfectly soft body being homogeneous^ k had the same value at all its points. (') W. Thomson, On the theory of magnetic induction in crystalline and non-crystalline substances {Philosophical Magazine, 4' series, t. I, p. 177-186, i85i. - Thomson's Beprint of papers on electrostatics and magnetism, art. XXX). Il8 BOOK VIII. - THE THEORY OF FISH. The equations (i) can be written in a more explicit form. Let XD be the magnetic potential function of the perfectly soft body. Let ^ be the magnetic potential function of permanent magnets. We will have ^; = u + ijj) and the equalities (i) can be written \dx dx I Permanent magnets are assumed to be fully defined. We can therefore consider their magnetic potential function ^ as known. The equations (2) show us that, to determine the magnetization at each point of the perfectly soft body, it is sufficient to determine the function TD. Here is the fundamental transformation on which the determination of this function is based. Let's differentiate the first equality (2) with respect to x, the second with respect to j>/" and let's add the obtained results member by member. We find Ox oy 03 But, in the first place, any point (.r, jk, ^) of the perfectly soft body being outside the permanent magnets, we have, at such a point A\tp = G. Secondly, at any point of the soft body, we have [Book VII, Chap. III, equality (8)] The previous equality is therefore reduced to (4) (i + 4TrA-)AX9 =0. If the body is magnetic or paramagnetic, k being positive, it CONDITIONS OF EQUILIBRIUM. "9 The same is true for (i -h 4'^^)- For diamagnetic bodies, according to the definition of Sir W. Thomson, A' would be negative; but for all known diamagnetic bodies, A" would have a negative value. Thomson, A' would be negative; but, for all known diamagnetic bodies, A" would have a value very small, much less than -- - We will therefore not exclude any known body, restricting ourselves to the study of bodies that satisfy the following condition: The quantity (i + 4 "A) is positive. We see then that the equality (4) can be written (5) Ai:)=o, which transforms equality (3) into (fi) ùx -ùy ùz According to the definition given in Book VIT, Chap. IV, § 1, this last equality is equivalent to the following theorem: U according to Poisson's theory, the magnetization of a perfectly soft homogeneous body is always a solenoidal magnetization. According to what we have seen in Book VII, Chapter VII, § 2, this distribution is also a simple lamellar distribution. Equality (5) shows us that the magnetic potential function "O of the perfectly soft body is harmonic at any point inside this body. Consider a point on the surface of the perfectly soft body. LetN be the normal to the surface at this point towards the interior of the perfectly soft body. The equalities (2) will give us ..c".(N,...)" = -.(|".^), But, on the other hand, we have [Book VII, Chap. II, equality (9)] dX3 dXD 0^, d^, = 4Tr||t-l)C0s(N/, x] We have therefore, at any point of the surface of the body perfectly soft, (7) ^'^4^^)^-4-^H-4^A:^=o; 120 BOOK VIII. - L.V THEORY OF FISH. jrr can be considered as a known quantity, this equality ( 7 ) constitutes a relation between rrr- cl-z^rr- ^ ' ^ oN, c)Ne The function sought Ï3(^, y, z) is, from the above, subject to the following conditions: 1° It is continuous in all space; 2° It is harmonic lant outside and inside the body perfectly soft; 3° It behaves at infinity as a potential function; 4° At any point on the surface of the perfectly soft body, its first order partial derivatives verify the equality (7). Are these conditions sufficient to determine the function '0?J1 It is easy to see that two distinct functions t) and 'O' cannot verify all these conditions at the same time. Suppose, in fact, that two distinct functions XO and W verify both these conditions, and let us pose e = '0'- t). The function would be, like functions 13 and O', harmonic both inside and outside the perfectly soft body. If therefore we designate by dçi an element of volume inside the perfectly soft body and by dve an element of volume outside the same body, we will have (8) /'A0^Pc-t-(n-4-n:A:) f\&dvi = o. But Green's theorem gives us, by designating by S the surface of the perfectly soft body, On the other hand, the equalities (i + 4^A-)^.-f-^-+-4^/^^=o, , , , , dV dXD' , - dx^ from dB - ' O - = o. ôy ' dz CHAP. I. - CONDITIONS DK L'EQUILIBRK. 121 nioiîlrenl that we have, at any point of the surface S, J/equalilé (8) thus becomes , . . , , , . . . r . . àe d& d& The quantity (i + 4~'>^) is positive. The quantities -,-7 -^ jz are continuous at all points, both outside and inside the body perfectly smooth. The previous equality therefore requires that we have, in all points both external and internal to the body perfectly soft, from If we add that, like the functions t) and V, must be continuous in all space and equal to zero at infinity, we V!)it that we will have in all the space e = or The stated proposition is therefore proven. We thus see that the conditions previously indicated are sufficient to completely determine the solution of the problem of magnetization by influence, the existence of this solution being admitted. Does the problem in question always admit of a solution? This is a question that we shall reserve for the moment; we shall have occasion to examine it in Book IX in a more general form. 3. - Another way to equate the problem of influence magnetization. The equation of the magnetization problem, as just indicated, is due to G. KirclihofTC ). (') G. KiRCHHOFF, Ueber den inducirten Magnetismus eines unbegrenzten Cylinders von weichem Eisen, § 2 {Crelle's Journal, Bd. XLVIII; i853.- A'ïVc^hoff's Abliandlungen, p. 198 ). li'i. BOOK VIII. - THE THEORY OF FISH. The form in which Poisson (') equated the problem of magnetization by influence, a form used not only by Poisson, but also by F.-E. Neumann, Lipschilz, Béer and Cari Neumann, differs slightly from the previous form. We will indicate it here. Let 'Q be the magnetic potential function of the whole system, given by the equality \'; = t3 4- 1^. Let's put We will then have, instead of the equalities (i), the equalities (9) X^A-^, ..'o^-A-^, 3 = A-^-2, dx- ôy oz which show us that the magnetization of the perfectly soft body is determined when the function o is known. Now the function o is defined by the equality (10) çp _^ t) -t- \J? = o, in which the known function \^' and the unknown function V) appear. But this one is linked to the function o by a very simple relation. We know, in fact, that we have, in general [Book VH, Ghap. III, equality (3)], 0(^,j,.)=-§||.,l>cos(N,,^)||^-,^-y'||^ - dv, the first summation extending to all the elements of the surface of the perfectly soft body, and the second to all the elements of the volume of this body. In the present case, according to equality (6'), this equality reduces to i:)(a7,j',4;)=-g|l.l,cos(N,,^)|| (') Poisson, jWe'woï/e sur la théorie du magnétisme, § 2, n" 1Q { Mémoires de l'Académie des Sciences, années 1821 et 1822, t. V, p. 247). (^) One could just as easily try to determine the "C" function or the ID function as the 9 function. We have chosen the latter only to conform our formulas to those used by other authors, notably F.-E. Neumann {Vorlesungen iiber die Théorie des Magnetismus, namentlich liber die Théorie der magnetischea Induction; Leipzig, 1881). CHAP. I. - CONDITIONS OF EQUILIBRIUM. 123 By virtue of equalities (9), this last equality takes the form The comparison of (10) and (11) shows that the function (p) verifies the relation Now we shall prove that there cannot be more than one C3 function verifying this equality (12), so that this equality can be used to determine the distribution of the magnetism induced on the perfectly soft body. Suppose that two distinct functions, o and es', verify the equality (12), and let us pose ^ =: es'- C5. We will obviously have O oNi r This equality shows us that ^ could be considered as the ' ordinary potential function of an electric layer distributed on the surface of the soft body with a surface density ^ Mi The function .- Q"^ \L"'^ -^_Lr-^S = o. ^ d ^i Mm Let us take a point M infinitely close to the normal v/ passing through a point P of the surface S. Let us suppose that this point moves ( ' ) F.-E. Neumaxn, Vorlesungen iiber Théorie des Magnetismus, namentlich ûber die Théorie der magnetischen Induktion, p. 110; Leipzig, i88i. CHAP. I. - CONDITIONS OF BALANCE. infinitely little on this normal, towards the interior of the soft body, of a length (hi. Let us write the new equality analogous to equality (i2 bis), and subtract these two equalities from each other. We will find Let's make point M tend to point P and we will find at the limit Any equality of the l'orme (12 bis) determines a single function (p(M) when we replace \^ (M) by a certain potential function. Let us take for vg) (M) the potential function of a mass equal to to the unit placed at point P, i.e. the quantity -T=r- Let ^ ^ ^ PM G (M, P) the corresponding function es (M), a function determined through equality 14 ) TT-:^ -I- G (M, P) - A- V W^ ' '^:^^ d^ = O. Let's multiply the two members of equality (i4) pai" ) Let's multiply the two members of equality (i3) by -z=r-' Let us subtract member by member the obtained equalities. We will have ^ n pG(/n, P) dx?p(P) I do(m) d / 1 ]^s r^o Let dy be an element of the surface S surrounding the point P. Let us multiply the two members of the previous equality by dy^Bz). (a;2_(_j^2_j_^2)2 This formula shows us that a uniformly magnetized sphere exerts at any external point the same action as a magnetic element placed in its center, having for direction of axis the direction of magnetization and whose magnetic moment would have for components (4) A=^rR3oil,, B=i:iR3\)>,, G=|TrR3S. 3 o 3 D. - II. 9 l3o BOOK VIII. - THE THEORY OF FISH. If the point (o:, y, z) is inside the sphere, we have dV 4 av 4 (JV 4 T-=~5'"^' Tr=~T'^.Xi ':r-~i'^^ ax 3 ôy 6 "^ oz 3 and, therefore, according to formula (2), (5) '0(x,y, z)='*^'K(Xx-h^i\,y-i-ez). This formula will allow us to solve the problem of the magnetization of a homogeneous magnetic sphere in a uniform field. The magnetic potential function '^{x,y,z) of a uniform field is a linear function of the coordinates 5 either (6) ■^(x,y, z)^-{Fx-i-Gy^llz -^K); F, G, H will be the components of the magnetic action at any point of the field. Let's imagine that we impart to the sphere a uniform magnetization having as components X = ^,- F, *= -, G, G = ^ H. We would then have, according to the equalities (5), (6) and (7), dz Now, according to what we have shown in the previous chapter, CHAP. II. UNIFORM MAGNETIZATION IN A UNIFORM FIELD. l3l these last equations unambiguously determine the magnetization taken by a sphere whose magnetization coefficient is k in a field whose magnetic potential function is ^. This magnetization is therefore given by the equations (7). Thus, in a uniform field, a solid magnetic sphere magnetizes uniformly. U magnetization has the same direction as the action of the field and is proportional to the field strength. According to equality (3), the magnetic potential function of a sphere placed in a uniform magnetic field has the expression, outside this sphere, T.k (8) X){x,y, z)- R-i (Fa^-i-Gj 4-Hs). I -H q TT A: (a;2 -f- jK^ -H z'^y^ § 2 - Magnetization of a solid ellipsoid in a magnetic field uniform. The previous method also applies to the study of the magnetization taken by a homogeneous ellipsoid in a uniform magnetic field (*). Let \[x, y, z) be the ordinary potential function of the uniformly electrically filled ellipsoid with density i. Let us take the center of the ellipsoid as the origin and the axes of the ellipsoid as the coordinate axes. From what we have seen in studying the attraction of ellipsoids, we have [Book I, Chap. VI, equalities (i5)] dx dV dy dY_ dz = - ihx, = -ïMy, = - 2N3, L, M, N being [Book I, Chap. VI, equalities (i3)] three constants A, (' ) Poisson, Second Mémoire sur la théorie du magnétisme, § I {Mémoires de V Académie des Sciences, années 1821 et 1822, t. V, p. 488). l3'2 LIVnK VIII. - THE FISH TIIEOUIE. x, V for the interior points of the ellipsoid and three functions of X, y, z for the exterior points. Equality (2) then becomes (9) ■0(x,y, z)=-2{XLx-h\i\,My-h 3N2). From this equality, we deduce, for the interior points of the ellipsoid, (10) -T- =2<Â.-À, -T- =2l)b[Jl, -T- =2SV. ox oy oz With these results, let us place a homogeneous ellipsoid in a uniform magnetic field, defined by equality (6). Let us impart to this ellipsoid a uniform magnetization having for components F, I l -1- V! /i A I Ir (n) It is easy to see, from eq. (6) and (10), that this magnetization will verify the eq. dy and will represent, therefore, the equilibrium magnetic distribution taken by the ellipsoid. Thus, a homogeneous ellipsoid, placed in a uniform magnetic field, magnetizes uniformly; but, in general, the direction of V magnetization does not coincide with the direction of the field lines of force. Equalities (9) and (i i) give the following expression for the magnetic potential function of a homogeneous ellipsoid placed in CH.VP. II. - UNIFORM MAGNET IN A UNIFORM FIELD. l33 a uniform magnetic field g 3. - Determination of the magnetization coefficients. Poisson (*), to whom we owe the solution of the preceding problems, showed how the results obtained could lead to the determination of the constant k. Let us suppose that we place a homogeneous sphere made of the substance whose magnetization coefficient we want to determine in a uniform magnetic field known with precision, for example, in the earth's magnetic field. This sphere will magnetize according to the laws indicated in § 1. Let us take for x-axis the magnetic meridian directed to the north, for y-axis a horizontal line, normal to the previous one and directed to the east, for z-axis a vertical line directed to the zenith. Let T be the terrestrial magnetic force; let i be the inclination. We will have iF = T CCS i, G= o, H = - Tsini. Let R be the rajon of the sphere whose center is supposed to be placed at the origin of the coordinates; let (^, y, z) be a point outside the sphere. Let us assume According to equality (8), the magnetic potential function of the sphere at the point (x, j"', z) has the value Kk 3 R3 X)(x,y,z) = jT(xcosi - zsini). i + I^A: 'This sphere will exercise, on a magnetic pole equal to the unit (' ) Poisson, Mémoire sur la théorie du magnétisme, g III {Mémoires de l'Académie des Sciences, t. V, p. 247; years 1821 and 1822). l34 BOOK VIII. - THE THEORY OF FISH. placed at the point (^, y^ z) an action whose components will be [cost 3(xcosi - z^'mi)x~\ -. 7^ J' R3T (i4) ' H = 3(x cosi - z smi)y' sin t 8 (.2? CCS i - ^siiii) A magnetic pole equal to unity placed at the point (^, j>^, z) will undergo a total action whose components will be (i5) Y --. G H- H, Z ^H-j-Z. These formulas allow us to know easily the direction that a small magnetized needle placed at the point (x,y,z) will take. If, for example, this needle is mobile around a vertical axis, its magnetic axis will be oriented in such a way as to make an angle u with the direction of the a-axis, counted positively towards the east, and we have G-t-H tang'J = -pr;^2' or, by virtue of the equalities (i3), (i4) and (i5), (xcosi - zsini)v ^i"- ;:. (i6) tangu with (i6 bis) cosi(' - [0.) -f- 3 [J. {X ces i - Z sin i) X r2 R3 i-H i ttX: Let's assume that the needle can move in the horizontal plane that contains the center of the sphere. We will then have 2 = 0, and the CHAP. II. - UNIFORM AIMING IN A UNIFORM FIELD. l35 mule (i6) will take the much simpler form tang'j = I - 1^ -4- 3 (^ Let cp be the angle of the OM direction with the x axis {Jig. 17) Fig. 17 B ' joining the center of the sphere to the middle of the needle BA. We will have X . y CCS o - -, sin " = " > ^ r ' r and the previous formula will become 3[Ji sinœ coso (17) tangu (I-'X) (^H-.^-liLcOS2çj According to this formula, the angle u is equal to o when the line joining the center of the sphere to the middle of the needle is parallel or perpendicular to the magnetic meridian. The angle u passes through a maximum at the moment when the angle o takes a value determined by the equality (18) COt^çp i -t- 2[JL The quantity a, defined by equality (16 bis), is positive and in , R' smaller than - * If the distance is very large compared to R, pi dt l36 BOOK VIII. - THE THEORY OF FISH. comes very small; equality (i8) reduces to COt^çp - I, and the maximum of u occurs approximately at the moment when the line OM is bisecting the angle XOY. To determine the constant k, we observe the value of u which corresponds to (p = - - Then we have 3 - I^ (19) tangu = (l-f.)(T-f-.^-^ This equality will make it possible to determine jx, and, consequently, according to equality (16 bis), the magnetization coefficient k. Experiments made by various experimenters on different species of soft iron have given as value of the quantity i + i^/c a number slightly lower than the unit. § 4 Determination of the inclination by measuring deviations horizontal. Formula (16) provides a means of determining magnetic inclination by observing the angle by which a needle, moving about a vertical axis, is deflected from the magnetic meridian by a sphere magnetized by the earth's magnetic field. The formula ( 1 6 ) can be written 3 fi. (xcosi - zëini)y i - fj. ;'2 CCS i tangu = or 3 |Ji (xcosi - zsmi)c[; ' ^^ 1 - [X r'^ CCS i tangu 3(JL xcosi - zsini y - a; tangu 1 - [j. r'^cosi We observe the value of the angle u for two positions of the needle, symmetrical with respect to the horizontal plane containing the i CIIAP. n. - UNIFORM AIMING IN A UNIFORM FIELD. iSy center of the sphere. If the first observed value, u, is given by the previous formula, the second, w, will be given by the formula lanfïtv 3[j. X cosi-{- z sini y - X tang w i - [x r'^ ces i The two formulas we have just obtained give tang'j tangw 6 a x -I - - , y - X tangu y - x latig w i - ;j. r^ taneu tan'^rw (y\x z y - a? tan g u y - ;iLaiigi^i^ i - [jl r We deduce from this tanjr"' tangu (20) tangi=- -^ H ./ h tanfffj' tansru y - rptangw y - a^tangu This formula will allow us to determine the magnetic inclination by observing only the deviations of a moving needle around a vertical axis. We are content to indicate here the principle of this ingenious method, which can be completed in such a way as to take into account the length of the needle and the magnetization induced in the sphere under the influence of this needle (' ). (' ) F.-E. Neu-MANN, Vorlesungen ûber die Theorie des Magnetismus , g 20; Leipzig, 1881. l38 BOOK VIII. - THE THEORY OF POISSOX. CHAPTER III. HOLLOW SPHERE IN A UNIFORM MAGNETIC FIELD. § 1 - Magnetization of a hollow sphere in a magnetic field uniform. The two examples we examined in the previous chapter were treated directly by taking as a starting point the equations which relate the components of the magnetization at a point on a perfectly soft body to the partial derivatives of the magnetic potential function. Such a direct method applies only to bodies which assume, in a uniform field, a uniform magnetization; the only known bodies which possess this property are the solid sphere and the solid ellipsoid. In all other cases, it will be necessary to take as a starting point the analytical methods that we have indicated in Chapter I. We will give an example of the application of these methods, by treating the magnetization taken in a uniform field by a mass of soft iron enclosed between two concentric spheres. This magnetization was determined by Poisson ('). According to equality (12) of Chapter I, the problem comes down to the search for a uniform, finite, continuous function 'f in all space and verifying, at any point of the magnetized body, the equality where ^ represents the magnetic potential function of (' ) Poisson, Mémoire sur la théorie du magnétisme, § III {Mémoires de l'Académie des Sciences, t. V, p. 247; years 1821 and 1822). CHAP. m. - HOLLOWING SPHERE IN A UNIFORM FIELD. I Sq acting masses. At any point of the soft iron mass, the components of the magnetization will be given [Chap. T, equalities (9)] by the equalities do .. d^ ^ do (2) &\0 ÔX Db at Oz A remark about equality (1). The surface S, which limits the spherical layer, consists here of two distinct surfaces {fig. 18): an outer sphere S(, of rajon R,, and a sphere Fig. 18. S2, of rajon R2. Let N be the exterior normal to the first sphere, and N2 the exterior normal to the second. Equality (1) should be written in the more explicit form (3) vj^ + ^ + ^C 1 /'i rfSi S do I s. ^N2 r^i dSi The function ^, corresponding to a uniform magnetic field, is of the form ^-^ (4) xj?) = - (Fa^ -t- GjK +- Hz + K), F, G, H being the constant components of the magnetic action at a point of the field. Let us take the origin of the coordinates at the common center of the two spheres Si, S2. Instead of the Cartesian coordinates x, y, z, let us take the polar coordinates p, 9, 'b (Jig. 19). We will have / 37- p sin({; cos6, (5) \ y - p sirnj; sinO, - p costj;. l4o BOOK VUI. - THE THEORY OF FISH. The expression (4) then becomes (4a) '^ - - p(Fsin4^cos6 -+- Gsirn}; sinO -}- H cos4/) - K, To transform equality (3), we will notice that, if we denote by R, Oi, '^i the coordinates of a point of the element dS^ Fig- '9 i/" -> and by Ro, O2} ^2 the coordinates of a point of the element "/Sa, we have (6) r^- Rf-i- p2- 2pRi [cos'\i cos sin(j;i ces (0 - 61)], ri - RI -h p- - 2pR2[cos4' cos({/2+ sii'î' sin');2 ces (6 - 63)] , oJS2= R| sint]^2 ddi d'\i2. To obtain all the dSi elements of the sphere S", we will have to vary 9, from o to 2tc and (j^) from o to u. These formulas will transform equality (3) into an equality that we will call (3 bis) and that it is useless to write. We will try to verify this equation (3a), which determines a single function (p, by an expression of the form / tp = p (M sinij; cosO 4- N sintj/ sinB -f- Pcos'l) (7) ( - (m sint]; cos6 -+- re sirn}; sin6 -H/>cost|/) -+■ Q, P M, N, P, m, n, p, Q being seven constants. If, by a suitable choice of these constants, we can bring the function cp given by this equality (7) to verify the equality (3 bis), we have ClIAP. III. - HOLLOW SPHERE IN A UNIFORM FIELD. l4l The C5 function we are looking for will certainly be determined within the soft iron. Equality (7) gives us (8) ip Let L be the geometric quantity which has as components (9) M --^-3-, N-^, Let La be the geometric quantity whose components are P _^ We will have Os, ^N, /-i Os, c^Na /-a cos (Li, R,) kU R? + p2- 2pRiCos(p, Ri)p cos (L2, R2) l^S, ri cos ^^* rR?-t- a2_ f/S, Let's put [R2+p2_2pR2C0S(p, RO]^ (p,Ri) = M, (p,L,) = u. Let us denote by p the angle that a plane led by the half-lines p, R, makes with a plane led by the half-lines p, L,. We will easily find that we have ..,s cos (Li, Ri ) 1 ' s. [Rf+p2_2pR,COS(p,R,)]2 ;r>.>r C^^ /''^ ( COS M COS'J -4- sill M sifl 'J COS)f) ) sifl U C?M <5?p =^A-RfLi / / 7 -o ^^ (R2-+-p2_2pR, COS")^ "in " cos/i du ~- i-A'Rj Li cos'j / ° (R2_(-p2_ apRiCosw)-^ l4i BOOK VIII. - THE THEORY OF FISH. An analogous transformation applies to the integral ku S ^^i(!-Ri) ^ as, ^^' [R|-t-p2- ■>.pR2COS(p,R2)]2 and gives us ()o I " , C\ ()o I Os, to^i n Os, dN, /■, = 2TrA- Rf Li cos(p,Li) / -j of ^^^^ \ L ° (Rf + p2 - 2pRiCosa)2 r" sinMcosM - R| Lj cos(p, L2) / -du ^° (R| + p2 - 2pR2COSM)2 Equality (3a) must be verified for the points of the spherical layer. For these points, we have R2 < p < Ri. We can then, by posing COSM = /, eflfect the integrations that appear in the equality !(io). We find k X -^ ~ dS, - A- V --'- - dS, Os. dNi ri SJ dNi r^ ^ = ^TrA pLiCos(p, L,) ^ L2COs(p,L2) - If we remember that the components of the geometric quantities L,, L2 are the quantities (9) and (9 bis), we find -^'-^î ["-^RT-(T)'("-i)]^'"*-'" [''-^-(7)'(-^)j-*i f CHAP. III. - HOLLOW SPHERE IN A UNIFORM FIELD. Equality (3a) then becomes 43 - p(Fsint{ cos6 -I- G sint}^ sin6 -f- H cos4' ,£, n Fsint]; cosô -I- G'sint|^ sinO -4- H cost}^ ,g^ ) -x,\^ 7, ^s^j Let T be the magnetic action at a point of the field, whose magnitude is given by (i5) T =(F2+G2^-H2)2. The previous equality can be written t) = kT\ A.^ c^^(f^^ .^S, ( ^ [R?+p2-2pR,cos(p,R)]2 . n cosCR. T) ,_ - A2 V jdSi [Rl + P''-2pR2Cos(p, R)]î A transformation similar to that employed in D. - II. 10 l46 LlVnK Vlll. - L.V FISH THEORY. previous paragraph allows to write t) =27:A-Tcos(p, T) R?Ai/ L J, (RÎ + ?2-2pRi ;A2 / -^ , J, (Rl+p^-'^pR2COSa)-2^"J - of the j cosup (16) ^ - R^ To further the calculation, it is necessary to distinguish the points located in the cavity enclosed by the spherical layer from those located in the unlimited space. If the point (p, G, ^) is located in unbounded space, we have R2di>, c= Ce dv. If the body is a perfectly soft body magnetized by influence, we have [Book VIII, Ghap. I, equality (9)] X - k-~, i)b = A- -i- " S = A: -t^ ox oy oz and, therefore, (,) A = ./^-|*. B = ./|... C==A-/g*. Green's identity allows us to transform these expressions. If U and V are two regular functions in the considered space, if dS is an element of the surface which limits this space, we have dY /-v* = -/||Si|..-Su .N, ''S Let's do, in this equality, V = X, V = C5 and notice that the function (^ is harmonic inside the magnetized body, we will have, denoting by |, v), ^ the coordinates of a point of the <^S element, and, therefore, (' ) F.-E. Neumann, Vorlesungen iiber die Theorie des Magnetismus, § 43; Leipzig, 1881. CIIAP. IV. - BEER'S METHOD. l53 Let us now return to the equation [Book VIII, Chap. t, equality (12) which takes place at any point (.r, jk, s) inside the body. In this equality, ^ is the magnetic potential function of the masses outside the magnet, /- is the distance from the point (^, y, z) to the point (Ç, 71, "Ç) of the element dS. From this equality, we deduce dx r)cp dx 00 r k S ^^ ~- éS dN; dx or which, by a change in the order of the integrations, can be written /^*-/â*-^s dS = o. But r dr dx Oç so that the previous equality can be written V being the ordinary potential function of a homogeneous mass having the same shape as the magnet. In the case where the magnet has the shape of an ellipsoid, this equality is simplified. In this case, in fact, we have [Book 1, Chap. VI, equalities (i5)] bones i54 BOOK wine. - THE THEORY OF FISH. L, M, N being [Book I, Chap. VI, equals (i3)] three constants "k, [j., V, for the interior points of the ellipsoid, and three functions of ^,y, ^, for the exterior points. The previous equality then becomes J dx J dx O d'^i By virtue of equalities (i) and (2), this equality gives us the first of three relations It-^ (3) I -r- 2 X- X B = - -J^ f^ d., 1 -T- -J. A (J. J Oy ôz di> The last two are obtained in a similar manner. These equalities immediately provide the components of the magnetic moment taken by a soft iron ellipsoid in a given field. In the case where the ellipsoid is reduced to a sphere, we have 2 and the previous equalities become A = ()X (4) I 'Of i + -^.k G = 4 _ ,. J i + 3^A Oz dv, dv. dv. 2. - Beer's method. The works mentioned in the previous paragraph have provided the solution of the problem of influence magnetization only for a very limited number of cases. i r.liW. IV. - BEER METHOD. 1 55 Beer(') gave a method which allows to solve the problem of influence magnetization for any magnet terminated by a non-biased second rank surface. Beer had merely stated his algorithm. Dr. Cari Neumann (2), whose research on the arithmetic mean method was prompted by Béer's Note, demonstrated the legitimacy of the algorithm created by this physicist. Béer's method would, in most cases, be rendered inapplicable by the length of the calculations. But it has at least the advantage of proving, for a very large class of bodies, the existence of a solution to the problem of magnetization by influence. We have seen [Chap. I, equality (12)] that the difficulty of the problem of magnetization by influence consisted essentially in finding a function being the potential function that defines the magnetic field. By means of the function ^^, let us form the series of functions (6) If the boundary surface of the magnetic body is a non-biased second rank surface, we know (Book II, Chap. IX, § 5) that the series will be uniformly convergent. For a magnetic body or (') A. Béer, AUgemeine Méthode zur Bestimmung der elektrischen und magnetischen Induktion {Poggendorff's Annalen, Bd. XLVIII, p. 187; i856). - Einleitung in die Elektrostatik , die Lehre von Magnetismus und die Elektrodynamik, p. i58 ff; Brunswick, i865. (") C. Neumaxn, Untersuchungen iiber das logaritlimische und Newton'sche Potential, Ch. VI; Leipzig, 1877. l66 BOOK VIII. - THE THEORY OF FISH. paramagnetic, but not diamagnetic, the quantity of the is a positive number and less than i . Jja series |_ ï-\-\'Kk \i-t-47rA:/ \i-\-^-Kk/ J will also be a convergent series. But, according to Green's identity, we have ,,",._,,,. S ^'I. s, Let's multiply the two members of the first of these equalities by - 7-: the two members of the second by ( -, - 7 ); - - - and add them member to member. We will find i-)-4tiA:^ \-\- \T.kVJ d'^i r If therefore we pose 1 -t- 4^" the function co, determined by equalities (7) and (8), will verify equality (5) at any point inside the magnetized body. The function cp being thus determined at any point of the magnetized body, the components of the magnetization will be given by the equalities ox oy oz For points outside the magnetized body, the function cp will be determined by the equality (5) CHAP. IV. - BEER'S METHOD. iSy Instead of determining the function 'j, we can determine the magnetic potential function V) of the magnetized body. The definition of the function o gives The previous equality becomes O tohi r By virtue of equalities (7) and (8), this last formula becomes ( N ,., _ A- o fà^ ^ fjizk dw / f^r.k yd^>" ■^l The fictitious layer equivalent to the magnetized body has therefore the following surface density (10) !J = \'dX^) ^ 4-Â- d^' ( 4 -A- \^dW Béer's method applies, whatever the positive value of k, to any magnetic body bounded by a non-biased surface of second rank. M. Cari Neumann showed that, given a magnetic body bounded by any surface, one could assign a positive number K, depending on the shape of the surface, such that Béer's method applies to this magnetic body for any value of k less than K. BOOK IX. INFLUENCE MAGNETIZATION AND THERMODYNAMICS. CHAPTER ONE. THE THERMODYNAMIC POTENTIAL OF A MAGNETIZED SYSTEM. 1. - Defects of the previous theory. In the preceding book, we have just explained some of the analytical developments to which the theory of magnetization by influence devised by Poisson gives rise. The consequences of this theory do not agree in every respect with experience. Let us take the very equations on which this whole theory is based [Book VIII, Chap. I, equations (2)]. al,,, 1)1, S being the components of the magnetization at a point of the perfectly soft body, xg> the magnetic potential function of this field, XD the magnetic potential function of the soft body, k its magnetization coefficient, these equations are °^=-K^ (0 1)1 dx dz ]' Let's multiply by an arbitrary constant X the components of the l6o BOOK IX. - THERMODYNAMIC AIMANTATION. field at each point, i.e. the quantities -t- > -r- ? - -; let us also multiply by the same constant \ the components X, i)b, G of the magnetization; this last operation will have for effect to multiply by X the value of XD in each point, and, consequently, the values at each point of -r- ? ■--, -r-* bi so equations (i) were verified before this operation, they are still verified after. Thus the following proposition is proved: According to Poisson's theory, if a perfectly soft body with a given deformation occupies a given position in a field whose lines of force have a given shape, its magnetization varies proportionally to the field strength. Experience shows, on the contrary, that when the field strength increases beyond any limit, the magnetization of soft iron tends towards a limit state which is called the saturation state. The contradiction that exists between Poisson's theory and experience can also be seen in the following way, which is just another form of the previous one. Let us suppose that the magnetization coefficient k of soft iron is determined by the method previously indicated [Book VIII, Ghap. II, § 3]. Instead of finding, as the value of A", a number independent of the magnetic field employed in the magnetization of the soft iron, one finds a number which varies with this field and which is smaller the more intense the field. The equations (i) which represent, in Poisson's theory, the conditions of magnetic equilibrium, cannot be preserved, at least in general. G. Kirchhoff (*), in a remark at the end of his beautiful Memoir on the magnetization of an indefinite cylinder, has shown how these equations can be modified. For the constant magnetization coefficient k, he proposed to substitute a magnetizing function F(OIL) variable with the intensity 31L of the ai ( ' ) Kirchhoff, Ueber den inducirten Magnetismus eines unbegrenzten Cylinders von weichem Eisen. Anhaiig {Crelle's Journal, Bd. XLVIII, p. 348; i854. - Kirchhoff's gesammelte Abhandlungen, p. 217). CHAP. I. - THE THERMODYNAMIC POTENTIAL of aIMANTS. i6[ manlalion . The equalities ( i ) are then replaced by the following : The body is magnetic or paramagnetic if the function F(3'TL) is positive; diamagnetic, if the function F(3TL) is negative; G. Kirchhoff has shown how, once this starting point has been given, one can reduce the problem of magnetization by influence to the integration of a partial differential equation. We shall examine this reduction in a later chapter, and we shall see that the equations given by G. Kirchhoff are no longer, like the Poisson equations, at odds with experience. The equations (2) are introduced from the start by G. Kirchhoff as a first hypothesis which is not justified by any preliminary consideration; but they are complex enough for this hypothesis, which nothing prepares, to put off the mind somewhat. On the other hand, it is very difficult to find experimental verifications that are not very indirect and roundabout. It would therefore be desirable to deduce the equations (2) from some simpler hypotheses; this is what Thermodynamics (^) will allow us to do; it will also provide us with the explanation of a great number of phenomena which are directly or indirectly related to the study of magnetization by influence. We will therefore, with the help of Thermodynamics, resume the study of magnetized systems from its first principles. § "2 The internal thermodynamic potential of a magnetized system. A magnetized body is defined by the knowledge of the value and the direction that takes, in each point of this body, a certain (') P. DuiiEM, Théorie nouvelle de l'aimantation par influence, fondée sur la Thermodynamique {Annales de la Faculté des Sciences de Toulouse, t. II, 1888). D. - II. II l62 BOOK I. - MAGNETIZATION AND THERMODYNAMICS. This is a geometrical quantity that we callV intensity of magnetization at this point. Let us consider a system of isotropic or non-isotropic bodies which can be electrified and magnetized and let us propose to determine the form of the internal thermodynamic potential of this system, potential which we will designate by the letter ^. We have obtained [Book IV, Ghap. II, equality (i5)] the expression of this potential by assuming the intensity of magnetization equal to o at any point. We found that we had, in this case, .f ^ E(r - TS)-f-W-4-y ery, E being the mechanical equivalent of heat; r the internal energy of the demagnetized and demagnetized system; S the entropy of the system under the same conditions; T the absolute temperature ; W the electrostatic potential ; q is the charge at a point; a quantity that depends on the nature of the demagnetized system present around this point; finally the sign N indicating a summation which extends to all the charges of the system. The expression of the internal thermodynamic potential of any magnetized system must, according to what we have just said, be of the following form (i) # = E(r-T2)-HW-(-2e^ + .f', rf being a certain quantity which reduces to o if the magnetization intensity becomes equal to o at any point. It is the form of this quantity # that we have to determine. Let us divide the system into an unlimited number n of infinitely small elements, of any shape, which we will designate by dç,, dv-i, ..., dç,i- Let us keep these various elements infinitely far from each other. The system being in this state, the internal thermodynamic potential can be taken as equal to the sum of the internal thermodynamic potentials of each of the elements d^^^, dç^, .... dç,i considered in isolation. If we designate these last potentials CHAP. I. - THE THERMODYNAMIC POTENTIAL of MAGNETS. l63 tiels by #,, ^2> - ■ -i ^/o t^ovls will have, when the system is in this state, (2) § = ^i-h^2^...+ ^n Let OÏL,, OTlo, ---> ^^n be the values of the magnetization intensity at a point of the elements d^i, dv^^^ ..., dvn- We can write ^2 = ?2-+-^2, > 3) 9 II - 9/i+ ^'") (p,, çpa, ..., "", not depending on OÏL,, Dli'2, ..., OIL^, while ^'j becomes equal to o together with STL,, §'^ together with JTLo, ..., ^,j together with OTU,,. Now the determination of the thermodynamic potential of a non-magnetized system is based on the assumption (Book IV, Ghap. II, § 1) that, in the considered state of the system. E(r - TS) + w-i-y eg- = (fi ?"- The equalities (i), (2), (3), joined to this last equality, show that, when the various elements of the system are infinitely distant from each other, we have ^'=^; + ^'2 + ...-4-^;. So, in general, we have, (4) i'=^i\-^t^..-+in-^P'. it" depending on the arrangement of the various elements of the system in relation to each other and becoming equal to o when the whole system is demagnetized. We can always write, at least in one way, (5) r=§\ + §\-^...^ra, i\ becoming equal to o at the same time as OIL^, §\ at the same time as OKo, . . ., #^ at the same time as ^". We shall suppose that, among the various forms that can be given to a similar equality, there is one that verifies a hypothesis that we shall state. 164 BOOK IX. - AIMANTATION AND THERMODYNAMIQt'E. For the convenience of language, in the statement of this hypothesis, we will name the quantity J^j magnetic potential of the system on the element dvt, the quantity c^'^ magnetic potential of the system on the element dVi-, Let us imagine that, on the element c?^, we place positive or negative quantities p.,, p.',, {ji.'(, ... of a fictitious fluid [magnetic fluid), so that the following conditions are verified: 1° We have tjti -1- [J.J -4- ijl'( -+-...= o. 1° If {x^y, z), {x',y, ;'), (^",y, z"), ... are the positions of the masses u.^, a'j, a'|, ..., we have [Xi a? -+- ix\ x' -I- ;ji 1 x" -\- . .= -.U, dv\ , |Xi s + [ji'i z' -h ^,2 depending only on the mass p.2 and the respective location of the two points M,, Mo. By a demonstration similar to the one we used to establish equality (8), we will prove that "^,2 is proportional to jjl2. Moreover, the respective situation of the two points M,, M2 depends only on their distance ri 2. We will thus have an equality of the form (12) Xl2= Î^2^(ri2). Among the fictitious distributions equivalent to the element dv2^ let us consider one analogous to the one to which equality (6) is related r CHAP. I. - THE THERMODYNAMIC POTENTIAL OF MAGNETS. 167 porle. The point Ma has coordinates œ-i, y-2, ^2 1 the point M'" has coordinates dx" x'^= xj,-v- ^^-INJjM'a, So if we have we will have 72=72 dU M, m: dz. dl - M2 M'2 . MiIVl2=: r,,, M.M; = r;,^r,,+ -^-^ + ldr\4) \'^i = j^ ^?(^12) ^ dx^ here .Iv A^n 0X3 dx,i dvii-T dVn Equalities (5), (9), and (r4) leave only the function -S( ) a point of this element; dv-i a volume element of the magnet Ao ; (^25 JK21 ^2) a point of this element; X}^ the magnetic potential function of the magnet A, ; Oo the magnetic potential function of the magnet Ao. It is easy to see that the variable part of ^" reduces to - / Jv>i di>i-+ so that this quantity can serve as a potential for the mutual actions of the two magnets. We have also identically 1} dxt âXi l di>2, so that the mutual potential of the two magnets A,, A2 can take one of two equivalent forms çp= rilx,^ d,,= f (^, r, z) at the point (^, JK, -) can be written (i5) \'>(.r,jK,z) dil ^^--^! dv, (i, Yl, J^) being a point of the element dv, (A^iih, G) the magnetization at this point and the summation extending to all the elements of the system volume. It will be identical to the magnetic potential function studied in Book VII, Chapter III. Equalities (5) and (9) will give (.6) -''--fh? ■2 J \\ dx dv. (a^,jK, s) being a point of the element dv and (.JL, iftj, G) the magnetization at that point; the integration extends to the whole system. This function rî" does not differ from the magnetic potential ^ defined and studied in Book VII, Chapter III. From now on, we will replace the symbol j" by the symbol ^. The results we have just obtained are sufficient, as we have already pointed out earlier, to make it possible to take up all the developments which form Book VII; but, to push further, we must determine the form of the quantities -T',, i'.^^ . , "5'^^. rf', depends on all the parameters that define the element dv^ considered in isolation. This element, in the most general case, is not isotropic; the various directions that can be traced on it can be distinguished from each other; let us assume that they are related to a certain reference trihedron that is invariably linked to the substance of the element, the trihedron of the principal axes of dilatation. The dvi element is defined by 1° Its volume c/p, ; 2° The shape of its surface and its orientation with respect to the axes of elasticity; 3° A number of parameters a,, p,, ... (its density, its temperature, for example), which fix its physical and chemical state; 4° The components ^,, S,, C, of the magnetization along the three main axes of expansion; 5° The state of eleclrisation; lyo BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. With respect to this last class of parameters, we will admit, as we have already done in the study of electricity, that in order to define the state of the element dv^^ it is sufficient to know the total charge q^ of electricity that it carries, without the need to know the distribution of this charge. Having made this assumption, we will first demonstrate that the quantity §\ is proportional to dv\ and that it does not depend on the shape of the surface that bounds the element nor on its orientation with respect to the coordinate axes. Since the distribution of the charge q^ carried by the element dv^, does not affect the value of §\^ we can, to prove the previous proposition, assume this charge to be uniformly distributed inside the element o^c, with a solid density p,. Let's choose once and for all a cube-type, whose dimensions are infinitely small compared to those of the elements dv\, dv2i - - - 1 ^^/f Let's divide the element dçf into cubes equal to the cube tjpe, having all their edges parallel to the main axes of dilatation The element dvy will contain an unlimited number k of similar cubes. Let -i = - - > 1 2X1= X2 Now Ào does not differ from )v, because all the parameters on which the quantity depends have the same value for the element dvi and for the element dv2- We have thus which teaches us that the quantity ). has the same value for the element dv^ which contains the charge ^, and for the element dv^ which contains no charge. Thus, as we announced, the quantity )v, does not depend on the charge qi distributed on the element c^t^i. We can therefore, highlighting the parameters of which depends on X, , write X,= (J(SVi,l3 "Ci, a" p" ...) and, therefore, by virtue of equality (17), (-8) J;=:g(5l, J3, C, a,,p,. ...)dv,. 17* BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. This equality (i8), together with equalities (i), (4) and (i6), determines the form of the internal thermodynamic potential of an electrified and magnetized system. This form is the following (19) .? = E(r - TS) + W +^Bg + ^+Jq(3K, U, €, a, p, . . .) di'. ^ is the magnetic potential. The integration extends to all magnets. One must remember that the quantity Ç does not depend on the electric charges distributed on the magnets. § 3 - Hemimorphic, holomorphic, isotropic bodies. The function Çi{^, |15, C), in the expression of which we will imply the parameters a, ^, . must become equal to o when the magnetization becomes equal to o, i.e. when we have 5V = o, 3J = u, C=o. We can therefore write (20)! -+-cpii(5V, 13, €)5V2+tp22(^, Î3, €)l32-f-cp33(5V, îl, C)C2 ( +2023(5V, 13, C)î3(!l-t-2c?3i(5^, B, €)C5l + 2cpi2(5l, % C)5VI9, \ [Ji, V being three constants, and the quantities cp^^ not increasing beyond any limit when %, M, C tend to o. But this general form of Çj {^, M, C) undergoes, in most interesting cases, notable simplifications. It remains expressed by equality (20) only for substances for which one can distinguish, at each point, a direction from the opposite direction. If a crystal is formed from such a substance, it will be devoid of center. It will present this particular hemihedral which tourmaline, calamine, ... present and which we will name Vhemimorphy{^). It is therefore only for hemimorphic substances that the function Ç(J3l, IP, C) will be given by the formula (20). (' ) We understand here by hemimetry any hemihedral which deprives the crystal of its center; crystallographers generally give this word a more restricted meaning. CHAP. I. - THE THERMODYNAMIC POTENTIAL of aIMANTS. lyS Let us now take a holomorphic substance, that is, a substance such that at each point one direction and the opposite direction are equivalent. If such a substance forms a crystal, this crystal will have a center. This case is, from the point of view of the geometer, a special case; but almost all known substances fall into this special case. The function (j'(/3l, IP, C) shall not change if we replace the magnetization by an equal magnetization, but oriented in the opposite direction, i.e. if we replace respectively /SI, %, C by - ^, - lu, - C This condition requires that we have X = O, }X = O, V = o, so that, for holomorphic substances, ^(J^t, jD, C) is given by the following formula: i q{%, U, C) = oh(51, B, (ir)5^2-^ c?23(^, 13, €)B2+ 033(3, B, "t)C2 ^^''^i -+-2923(51, Î3, C)l3C-f-2?3i(^, B, C)(!:5H-2çi2(5V, 13, CHAPTER II. THE EQUATIONS OF MAGNETIC EQUILIBRIUM. g 1. - Fundamental equations of influence magnetization. In this chapter, we will try to determine the magnitude and direction of the magnetization at each point of a body devoid of coercive force, or perfectly soft, subjected to the action of permanent magnets. Let us begin by defining these two words in a precise manner. Let us consider a system containing a body susceptible to magnetization and let us look for the conditions under which the uncompensated work, carried out in any virtual isothermal modification of this system, will be zero or negative; among these conditions, which are the conditions of equilibrium provided by Thermodynamics, we will find some which express the following proposition: The magnetization has, at each point of the body considered, a certain magnitude and a certain direction. If, at each point of the body, the magnetization has this direction and this magnitude, this magnetization can no longer undergo any variation; the magnetic equilibrium is absolutely established. But, a body being placed in certain conditions, it may happen that the magnetic equilibrium is established on this body only after a more or less long time; it may happen that, for certain bodies, this time is extremely long, so that, during an appreciable time, the magnetization of such bodies remains appreciably independent of the conditions in which they are placed. 176 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. they are placed. This leads to the definition of two limit forms of magnetic bodies. We will say that a body is perfectly soft or that it is devoid of coercive force, if, in any circumstance, the magnetization in each of the points of this body takes, after a very short time, the magnitude and the direction indicated by Thermodynamics, On the contrary, we will say that a magnet is permanent if the magnetization in each point keeps an invariable magnitude and direction in any circumstance that this magnet is placed. Permanent magnets and perfectly soft bodies form the two extreme limits of the series of magnetic bodies. It goes without saying that all the magnetic bodies presented to us by nature fall between these two limits, without ever completely realizing either of them. Nevertheless, these two limits correspond to the only cases whose theoretical study can be made completely; we can study completely the permanent magnets because their magnetic state can be supposed to be given arbitrarily, and the perfectly soft bodies because this state is defined at each moment by the propositions of Thermodynamics. Consider a system of permanent magnets and bodies without coercive force. At a point (^,, j-,, s,) of the latter, the magnetization has as components A,i, ^S\y^^ G,. One can imagine that, without changing the position, the volume, the shape, the electrification, the physical or chemical state of the various bodies which constitute the system, one makes, within an element dvi, vary JU,, ^S^)^, 3), of arbitrary infinitely small quantities oJl9^, ûiib", ô3,. Equilibrium will be established on the perfectly soft body, if this isothermal variation does not cause, for the system, any uncompensated work. Since all the bodies in the system remain invariant in volume, shape and position, the external forces applied to the system do no work. The uncompensated work done is then reduced to the sign-changed variation of the internal thermodynamic potential. The latter is provided by equality (24) of the previous chapter j Cn.\l>. II. - LKS EQUATIONS OF EQUILIBRIUM. I77 it has the value, keeping the notations of this Chapter, (0 § = E(V - T2) -^ W -^2®' ^ -T-^ T-'^C^l^, ",?,-- )dv. Let us look for the variation undergone by this quantity when, within the elétnent I,, <.l.,, i)î),, 3, vary respectively by 8x,, Sill",, 88,, all the other parameters which fix the state of the system remaining invariable. The quantity E(r - Ts) + w+Ve'7 undergoes, under these conditions, no variation. Let dvi^ dv-i^ ..., dva be the elements into which the system is decomposed; let 'C' be the magnetic potential function at the point (^,, j^,, ^,) of the element f/c,; let V.^ be the magnetic potential function at the point i^x^^yi^ ;;2) of the element 0^(^25 ---- According to equality (16) of the previous chapter, we have ^-\ dx. dvi îHoi dxi dvo J\sn âXn dv" We have. therefore ('i) >,v ' ('^'^7,^ _u^'^>;^.i. _^-^^''';?-^ X \ Oxi dfi àzi ) Ox, dvi ^looO Ox, dv But we can write the following equalities vlo" - - Ox,, dv ,.), (>\'>, à Oxi Oxi 0<', Oxi ~ àxi 0^ 0-^ ^^-'^ Ox, dv, - -1.3 /■'-' Ox, ■M Ox, dvi M - Ox, dv-i - . -I dV: ^l." '-u, Oxn dx,, dv, Ox,, dXn \ D. .1 Oxi dvi ".ta - OXî dv,^. l"-i OXn dv"-< j. 178 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. These equalities give 0- - = 0, dx2 tox-2 0-Jv>i dx di>i = ÔXi ôJ OÂoi dxo dvi, dXn dXa We deduce oJl,i dxi di>i dxi OcAoi r.l)lO dxi dvi .l,,8 .d-Ç, atxi dv=, cwo dxi oX toxn. dvi dvi +...-I XnO atx,i OXi àXn dVn oX di>i àXn dvi dvn - \ Sjlni dx ^^^ dv^ '"dF,. dVn dv^ = -. - o.ti-t- T-^ol)bi-i- --1 oS, \dv \dx^ dji dzi 7 Equality (2) then becomes (3) 5.5 = Finally we have Ojlai \dVi. dxi I 8y^(oiL,a,p,...)^.=^-^('^^',:-'^ ^V-^ox, But equality gives dDï^i D\il = X]-h^^^-i-ej WoAdI dvi. so that we can write <Â>i SoiUi CHAP. II. - EQUATIONS OF EQUILIBRIUM. 1 79 Equalities (i), (3), (4) lead to the equality 011 1 ÛLt ~Adi I ÔaAoi dvi. This quantity ô^f must be equal to o whatever the variations Za>{, ôi)b|, o2,. If therefore we pose I (5) F(OR,a,p, d.l(D\'^,ac, p, ■■■) dOÏL we must have, at all points of a mass devoid of coercive force and subject to magnetization X= -F(01L, a, !3, ...)^, (6) \ Olb-- F(01L,a, 3, These equations are the fundamental equations of magnetic equilibrium; they have the form admitted, as a hjpothesis, by G. KirchhofT (Chap. I, § 1). Before making any transformation to them, we shall deduce some important remarks. These equations give dx at dz CAB Db ~ e which means that V magnetization at a point of an isotropic mass without coercive force and the geometric quantity whose components are &0_ &Q dV dx dy dz are directed along the same line. This proposition is found in all the theories of magnetic induction proposed since Poisson; it is a more or less immediate consequence of the assumptions on which these theories are based. O BOOK IX. -MAGNETIZATION AND THERMODYNAMICS. The equations (6) still give d-çy /à-çy /d- - ° " ^ ^ ax dy dz are oriented in opposite directions; the body is then said to be magnetic if the magnetizing function is large, and paramagnetic if the magnetizing function is small. If, on the contrary, the magnetizing function was negative, the magnetic axis and the geometric quantity whose components are --, - > - - would be directed in the same direction, the body would be called diamagnetic. § 2 - The problem of magnetic induction can be reduced to the determination of the function "Q^Xyy, z). We shall now examine how, assuming the form of the magnetizing function F(01'L, a, P, ...) to be known, one could equate the problem of magnetic induction. The method to be followed has been indicated by G. Kirchhoff ('). (') G. KiRCiiiiOFF, Ueber den Magnetismus eines unbegrenzten Cylindcrs voit weichem Eisen (Crelle's Journal, t. XL VIII, p. 348; i8j4. - Kirchhojff's gesammelte Abhandlungen, p. igS). CHAP. II- - THE EQUILIBRITIES OF BALANCE. l8l Let us posit, as we have agreed [BookVII,Ghap.in, equality (19)], n^=(-^)^(^)^(^^y Equality (7) can be written <^'") [ F(.^.,t^...., ]'="^'' If the function F is known, this equation (7 bis) can be solved with respect to OÏL, and gives (8) OÏL =a(n\'^a, p, ...)- Now let's put (9) X (n t;), a, p, ...)=- F [^(ntp, a, p, . . .), a, p, . . .]. We see that, whenever the function F has a known form, the form of the function is determined. By means of equalities (8) and (9), equalities (6) become V"; ■ '\l!, = X(n\'>, a, p, ...)^\ \ G = X(nx:), a, p, ...);^ These equalities show that, when the form of the magnetizing function is known, it is sufficient to know the function 'Q to be able to determine the components of V magnetization at each point of the perfectly soft body. We noticed, a moment ago, that when the form of the magnetizing function F(01L, a, ^, . . .) was known, the form of the function (H''?, a, p, . . .) was known; conversely, if Von knows the form of the functionXiJl'Ç ., a, p, . . .), the form of the magnetizing function F(01L, a, ^8, . . .) can be determined. Indeed, the equations (10) allow to write 0iL2=X2(nx:^ a, i3, ...)nt?. If we know the form of the function À, we can solve l82 BOOK IX. - THERMODVNAMIC F.T. MAGNETIZATION. this equation with respect to n "Ç, which gives (II) n-C) = a'(orL., a, [3, ...). It is then easy to see that we have (I?-) FiDK, a, p., . . .) = n(x,y,jz). We have seen that the function "^Ç^, y, z) satisfies, at all points in space not located on the separating surface of two different magnetic bodies, or of a magnetic body and a non-magnetic medium, the following equation [Book VH, Chap. III, equality (8)], ^ ' ^ \ôx dy (Jz From this, we can deduce the form of the second order partial differential equation which is satisfied by t? at all points in space, except at the various points of the boundary surfaces. 1° In the part of space not occupied by the magnetic media, the preceding equation reduces to Laplace's equation (i4) ^■<> = o. 2° Inside the permanent magnets, the magnetization at each point is supposed to be known; so that the second member of equation (i3) is a known function of x, y, z. If we denote this function by - 4''^?(^7 JK, z), the function "Ç will verify, at any point inside the permanent magnets, the partial differential equation of second order (i5) A\')^-4-n:p(:r, 7, ^) = o. 3° Let us now consider a point inside one of the perfectly soft bodies. At this point, the components =.1., ilV, 3 of the magnetization are given by the equalities (lo). The first of these CIIAP. II. - EQUILIBRIUM EQUATIONS. equalities, differentiated with respect to x^ gives i83 dx [ axrn-C), a, p ) âac ()X(n\'>, g , ^, ...) ap irro ~^ " ûx .)0l dx the terms of the second line disappearing by themselves when substance But we have the substance is homogeneous. duo dx d-Ç d^-Q dp d'--Ç &0 d^O \ dx dx^ dy dxdy dz Oxùzj àUV dX Let's replace '^^^ by this value in the expression of -r-; for to the let's say -;-) -y; ; let us add these three quantities taking into account the relation (i3), and we will have [t - 4-À(n\'^a, 3, ...)]At;> - 2 of the. V (16) l\àx) 'àx^'^\dy) 'ôy^ '^ \dz } Oz'^ dt) d^:) an') d-<> d-<> d^-<' a\'> dp d'-X^ dz dx dz dx "^ dx dy dx dy dX(nV->, a. 3. ...) dx d<> 1 ax(n-c), a, p, ...) a^ dp OOL dx dx 1 aji dx dx If we remember the meaning of n\'^, we see that we have a second order partial differential equation, of known form when we know the function \ (Hxp, a, [3, . . . ), that the function P must verify at any point inside a perfectly soft body. Equations (i4)> (i5) and (16) thus represent the second order partial differential equations that the function p must verify in the various regions of space separated from each other by the boundary surfaces. § 4 - Boundary conditions satisfied by the function P^x,y, z). Let us find out what conditions the function 'C^(.r, y-, z) satisfies at a point on one of the boundary surfaces. '84 BOOK IX. - AIAIATNTATION AND THERMODYNAMICS. Suppose that a surface separates two media, at least one of which is magnetic. Let ]N( be the normal to the interior of the first medium and Na the normal to the exterior of the second medium. Let X,,i)b|, G( and Xa, i)î)2, ©2 be the components of the magnetization within each of the two media and in the vicinity of this point. We will have [Book VII, Chap. III, equality (9)] ^''^ 4^(1^ + IS) = Il -^^ '^'(^^' ^)""^ Il ^^^ ''''(^^^ "'^ll The surfaces that we may have to consider are of five different kinds: 1° Separation surface of a permanent magnet and a non-magnetic medium; 2" Separation area of two permanent magnets ; 3" Separation surface of a perfectly soft body and a non-magnetic medium; 4" Surface of separation of two substances without coercive force; 5° Separation surface of a substance devoid of coercive force and a permanent magnet. 1° ^ the separation surface of a permanent magnet and a non-magnetic medium, let us see what becomes of the relation (17). Let us assume that the index 1 refers to the magnet and the index 2 in the middle. Let's replace the symbols -r^-, -^ by the symbols, more often used in such cases, ^^- and -r^" Inside the non-magnetic medium, we have Jlo2=o, 1)1)2=0, ©2=0, while inside the magnet the quantities ^A,, i(î),, G, are supposed to be known in advance. The relation (17) thus becomes S [Xj y, s) being a function whose value is immediately deduced from the data of the problem. 2° Similarly, at the separation surface of two magnets, the i85 CHAP. II. - EQUILIBRIUM EQUATIONS. relation (17) becomes If {x, y, z), 52 (x, y, z) being two functions whose value is immediately deduced from the data of the problem. 3° At the surface of separation of a substance devoid of coercive force and a non-magnetic medium, let us examine what happens to the relation (17). Let us suppose that the index 1 refers to the magnetic substance and the index 2 to the non-magnetic medium. Let us replace the symbols N,, N2 by the symbols N/, N^. Inside the non-magnetic medium, we have ,^1,2=0, 1)1)2=0, Qi- o. On the contrary, inside the substance devoid of coercive force, X, il'oi, S, are given by the equalities (10). The equality (17) becomes therefore (20) [i + 47rX(IlX?,a, p, ...)] d^i ' dN, 4" Similarly, at the separation surface of two substances without coercive force, the relation (17) becomes {11) [i + 47rXi(n-C),aiPi,...)j||^+[i+4^>^2(n-C>,a2, !32, ...)]^ = o. 5° Finally, Cl the separation surface of a permanent magnet 1 and a body without coercive force 2, the relation (17) becomes ^^^^ ^ +[i + 4^>^2(n^?, a, p, ...)] ^ = - - Si{x,y, z), Si[x^ y, z) being a function whose value is immediately deduced from the data of the problem. If we add that at infinity we must have (23) dx Ç = o, dy = o, at we will have stated all the boundary conditions which complete the determination of the problem of influence magnetization. l8(> BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. § 5 - Poisson approximation. We have seen [Ghap. I, equality (23)] that we have .f(ort)=,m2cp(DiL), 'j(311L) does not grow beyond any limit when OÏL tends to o. For small values of DIL, we can write (24) J(.')1L)= OlUcpCo). Equality (5) of this Chapter c^.T(orc) dDïU then becomes F(DH)= - ' 20(0) For small values of magnetization, the magnetizing function takes a value Ji nie, independent of the magnetization intensity. Equalities (6) then become i ï ' ^^'' I tt/U, = - (25) I dt) -2C3(0) éZ These equalities agree with the equilibrium conditions given by Poisson's theory [Book VIII, Chap. I, equalities (i)], provided that one takes as the magnetization coefficient the quantity (26) k = 20(0) Thus the magnetization of weakly magnetized bodies is given by Poisson's theory. Equalities (24) and (26) show, moreover, that whenever Cn.VP. II. - I.ES EQUATIONS OF FQL'ILIBRE. that the Poisson theory is applicable, we have (27) :fiDXi)= -r' -1 /l This remark will be of frequent use to us. With this remark, the expression (i) of the internal thermodynamic potential of a system containing magnets becomes (2) .7 = E(r - T2}-f-W+> 0./-4-;J-+ 2: This expression of the internal thermodynamic potential is equivalent to the one that has often been introduced in the research on the properties of magnets by authors who have not usually defined its nature and justified its origin (' ). (') J. Stefan, Ueber die Gesetze der elektrodynamischen Induction {Sitzungsberichte der Akad. der Wissenschaften zu Wien, LXIV. 2' Ablheil, p.igS; 1871); Zur Théorie der magnetischen Krâfte {Ibid, LXIX, 2° Ablheil, p. i63; 1874); Ueber die Gesetze der magnetischen und elektrischen Krâfte in magnetischen und dielektrischen Medien, und ihre Beziehung zur Théorie des Lichtes {ibid., LXX, 2° Abtheil, p. 589; 1875). - W. Thomson, General probleni 0/ magnetic induction {Beprint of papers on electrostatic and magnetism, n<" 700 et seqq., 1872, 2" edition, p. 549). - See also the writings of Messrs. E. Betti, Korleweg, von Ilelmholtz, Boltzmann, Adler, G. Kirchhoff, Cari Neumann and E. Beltrami. l88 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. CHAPTER III. THE PROBLEM OF MAGNETIZATION BY INFLUENCE HAS ONE AND ONLY ONE SOLUTION. § 1 - Existence of a solution. We have seen, in § 1 of the previous chapter, that the 'problem of magnetization by influence' can be reduced to this one: To find a magnetic distribution which makes the quantity (I) V=V)-^ ǧ{m\,oi,^,...)dv, V integration extending to all magnetic substances. Does a similar distribution still exist? This is the question we propose to examine. We have [Book VII, Ghap. III, equality (17 his)\, the integration extending to the whole space. Equality (i) can thus be written .f'= /.fi(Ort,a, p, ..:)dv^ (2) '; -\- ^ Cu-Çdv,^ ^ jUt^dv, the first two integrals extending to all elements û?p, of the volume of the permanent magnets, the third integral extending to all elements dv-i of the space occupied by the non CIIAP. ni. - LX AND A SINGLE STATE OF EQUILIBRIUM. 189 magnetic, the fourth, finally, extends to all the elements di'^ of the volume of perfectly soft bodies. The first integral has a value independent of the distribution that affects the magnetism on perfectly soft bodies. Whatever the variations of this distribution, it keeps a constant value C. The second and third integrals have a value that can never become negative, whatever the distribution of magnetism on perfectly soft bodies. From equality (5) of Chapter II, which defines F(,')nL, a, |i. . .we deduce (3) .T(Dli,a,?, ...)= / ,. -dDK. Therefore, the quantity J(OTl, a, ^i, . . .) is always positive for magnetic bodies and always negative for cliamagnetic bodies. This circumstance does not allow us to predict the sign that the fourth integral in equality (2) takes for a diamagnetic body; but, for magnetic bodies, we can affirm that it is never negative, and, moreover, that it is only equal to o if the magnetization is equal to o at any point. Therefore, in the case where all the coercion-free substances in the system are magnetic bodies, equality (2) amounts to the following C being a quantity independent of the distribution that magnetism affects on substances devoid of çoercive force, and P a quantity that can never be negative, whatever this distribution is. This equality proves to us that the quantity §' is a quantity whose variations are inferiorly limited. From this, can we conclude that there is at least one magnetic distribution corresponding to a value of ^' that is smaller than all the others, and, therefore, to a stable equilibrium state? By doing so, we would only be following the path traced by Gauss to demonstrate that there is a state of electrical equilibrium igo iivui; IX. - magnetization and tiikumodvnamiql'e. on conducting bodies (Book II, Chap. II, § 2; Book III, Ghap. V, § 2). But we know that this deduction has a defect of rigor; for, from the fact that the variations of a quantity are inferiorly limited, it does not follow that this quantity has a minimum. It is therefore under a reservation similar to that which weighs on Dirichlet's principle that we shall state the following proposition: Any magnetic bodies being subjected to the action of any magnets, one can find on these bodies at least a magnetical distribution which satisfies the laws of magnetization by influence and which remains stable if one maintains invariable the position, the shape and the state of the various bodies of the system. §2.-11 There is only one solution to the problem of influence magnetization for magnetic bodies. - It corresponds to a stable magnetization. The equations of the problem of magnetization by influence simply express the equality at o of the first variation undergone by the internal thermodynamic potential S when, at each point of the substances devoid of coercive force which the system contains, one makes vary the components -l), ilij, © of the magnetization of arbitrary quantities ôn^l>, oil'o, oS. Can this equality at o of the first variation of § take place for several distinct magnetic distributions? When it does occur, is the function éf a minimum, such that the magnetic distribution is stable? These are the questions that we will now examine. The solution to these questions follows from the study of the second variation of rf , which we will first form Fexpression. Suppose that at each point {x,y,z) of the coercive force-free masses in the system, the components .^l), 1)1), 3 of the magnetization vary by oX,, h^S\>, o3, and let's say 8(î) = b ot, oC - - c o/. Cn.VP. in. - ONE AND ONLY ONE STATE OF EQUILIBRIUM. I9I ", b, c being three finite functions of x^ y, z, and ot an infinitely small quantity independent of x^ y, z. § experiences a first variation which can be written (4) Lf ^ o.y-i- 5 Ci{dXL, a, P, . . .)dv. If the magnetization had varied only in the c?p element, IJ would have undergone a variation oi^] if it had varied only in the dv-i^ element !J would have undergone a variation o-y'^^ . - and we have (5) oiT-oizr+o\^. o"?T, dVi^ dv2, - ■ ■ ■) d^n being the elements into which the masses devoid of coercive force decompose. Equality ^-J -\o dx dv -U .\o dx' dv', in which the first integration extends to all bodies devoid of coercive force and the second to all permanent magnets, gives us (6) '■s^=:Kt*-^- ¥;■"--"-§'=-)*■ -,l)iOi -T - \dvi-iaxi W oi Oi axi C/V2 Oi dx'. dvi +...dv',-\-... tox" dVr. dç\, dv'.,, . . . dv are the elements into which the permanent magnets are decomposed; 5, always designates a variation obtained by changing only 8jl>i, 8\fti, 83, the components of the magnetization in the element dv\. But calculations analogous to those which, in the preceding Chapter, were used to establish the equality (3), will prove to us: 192 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. 1° That we have -T- 0-_l,i + - - oljbi -4- ^-^ oSi ()xi dyi àzi ■+ di'i -I 1 8 ^"^^^ A 3 ^^'^" OX II ' 0, ^' C/\£>r, a'* That we have -l-l 0] -r- dxi c-the ' (Jo:-! -i,; dx 5.1,1 0M.M dxi ()x\ di'i dv-i dvy dv\ d.), -Xn dxi O-,!"! àXn M dxi à-7 a-A-M tox'n dvi dv'" dvx dv'p. The second member of this last equality can still be written, remembering that o-loi = ai 8/, Z^S\)l= bi^it, oCi = Ci 8/, --,:, "1 /"'-^ d,, àxt '^''dx\ a, !'''dv, dxi dXn à ! 6t. All these calculations transform equality (6) into 015" = -i,. âx" d - "1 -^; - ■ d^'^ âxi 4.,'. "dx' ^^-àr''-^ 'o/; 80^, - - - , o"-T Ofit analogous expressions. We can therefore CHAP. ill. - ONE AND A SKUL STATE OF BALANCE. replace equality (5) with the following '93 LJ = of ' '^'é-\ d-L dv.^-^ dL ÔX-i di'3-h. .+ Ô-L àXn dVn J dv\ "1 - - dvi - az-T- OX;, di>3 + . .+ ^ ''2' dxn dVnJ dvi. ÔXi dvi -) d ' âx.y di.2 -H . ..+ tox" dVn j dv'i Suppose that, on the perfectly soft body, we distribute a magnetization whose components at the point (.r,jK, ^) have values <7, 6, c. Let Q be the magnetic potential function of this distribution, a function defined by the equality (7) d r c'.r dv. It is easy to see that the previous equality can be written (8) 5:T= / .1, - di-^ / \X'~, dA^t. \J II dx / On the other hand, we have toO\<, oOli, or, by virtue of equality [Chapter II, equality (5)]. OR (9) F(OR, a, ,3, ...) 8.f(DR,a,p, ...) (^JfOlL. a, 3, ...) .OTL oOlI. F(Orc, a, p, ...) Therefore, by virtue of equalities (4), (8) and (9), one can write (10) dv' ) 0/, D. - II. ri t94 LIVRK IX. - MAGNETIZATION AND THERMODYNAMICS. Let's give again to JL, i)b, S the variations ^A, - a^t, 01)1)= è S/, o3 = c8f, and let us look for the variation ù^^ undergone by 5î. We will have (M) and 'S r\\. , , , , ; UHo' - -, \\di> - o J II"^37 II ^ r\ . à9.\ , ^ r\ dQ / X-T- di> = and I a -r J I <^^| J I ^^ rfi'. This last equality can be written, designating by du any element of the space outside the perfectly soft body (12) ^J Besides X +3c)2 - (ilbc - 36)2- (3a - A "C)2- (^1,6- ||i,")2 or ([-) m^ - a^- -^ b' -{- c- - - ^^ -- Therefore, equality (16) can be written ' . j o<2 / /* r I P3 = - - ( i UQdu+ \ WQdv 0]^lF(DlL,a,p, ...) (J.')1L (ill,c- 36)2-t-(3a- Xc)2+(.A.,è- ili,")2 ' [F(i)lL, a, p, ...)J2 OIV Let us discuss the sign of this quantity ô^cf, limiting ourselves to the case of magnetic bodies. The term that begins the second member of equality (18) is certainly positive or zero; it can never be negative. For magnetic bodies, the quantity F(3TL, a, [3, ...) is positive; in general, this quantity decreases as Oli- increases; we are then assured that the quantity S^j", given either by the equal dv. 196 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. lity (16), or by equality (18), is positive, unless one has, in every respect, " = 6 = c = 0. For some magnetic bodies such as soft iron, F (Dit, a, ^, ...) increases with OÏL for small values of 01i; but the increases of this quantity are not extremely large, and, as OTL has at the same time small values, the quantity ()F(01L, g, i3, ...) [F(J1L, a, p, ...)p D\V^ has a much lower value than ôk [ F(JlL,a%, ...) ] i-'-^b^'-^c^-), so that S^^jr is still positive. We can therefore say that for all co/inus magnetic bodies, h^^is always positive. Hence this first consequence: If magnetic bodies are placed in the presence of permanent magnets, any magnetic distribution spread over these bodies, according to the laws indicated in the previous chapter, corresponds to a stable state of equilibrium. Moreover, 8^,^ being always finite and positive, we see that there cannot exist two distinct magnetic distributions such that OrT = o. Therefore, on such magnetic bodies, there cannot exist more than one equilibrium distribution. CH4P. IV. - THEOREMS ON MAGNETIC BODIES. 197 CflÀPITRE lY. SOME THEOREMS ON THE MAGNETIZATION OF MAGNETIC BODIES. § 1 - Perfectly soft bodies have no magnetism remanent. Before going on to the study of magnetic bodies, we will deduce, for magnetic bodies, some consequences of the preceding propositions, and particularly of the last one: // there is only one magnetic distribution suitable for equilibrium on a perfectly soft magnetic body subjected to the action of permanent magnets. Let ^ be the magnetic potential function which defines the field in which the body is placed and ï!) the magnetic potential function of the magnetization distributed on this body. The equations of the magnetic equilibrium will be the following: Let us assume, in particular, that the perfectly soft body is placed in a magnetic field whose intensity is zero at any point. We will have identically igo BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. and the previous equations will become x= - F(orL) ~, ox Ift. = - F(DrL) v^, aty az Now observe: i*^ That the function X) becomes identically zero if we have, in all points, cAo = o, 'ift) = o, G = ; 2° That the function F(OrL) does not grow beyond any limit when OÏL tends to o. We will then see that the previous equations are satisfied if we have Jlu = o, \)î) = o, G ^ o. Since there is only one magnetic distribution satisfying these equations, this distribution is known. Thus: a perfectly soft magnetic body placed in a magnetic field whose intensity is zero at any point does not show any magnetization. In other words, a perfectly soft magnetic body has no remanent magnetism. 1% - Bodies that magnetize uniformly in a uniform field. It is of great importance, for the study of the magnetization by influence, to know the function F(D1L, a, p, ...) or, which amounts to the same thing (Cliap. Il, § 2), the function X(n'>, a, p, ...). We will see here on which principles one can base the determination. Let us take a homogeneous body, to which corresponds a magnetizing function F(OIL). Let us place it in a magnetic field where it CH\P. IV. - THEOREMS ON MAGNETIC BODIES. 199 is uniformly magnetized. The magnetization intensity has then a value M independent of .r, y^ z. The components A, i)î), S of the magnetization verify the equalities oy dy I - F(M) L F(M) is, like M, a quantity independent of .r, y, z. Now these equations are nothing else than the equations which define the equilibrium distribution taken, in I field considered, by a body of the same shape as the one we are studying and having a constant magnetization coefficient /<:, whose value would be precisely equal to F(M). If we remember that these last equations determine a unique distribution, we arrive at the following theorem: If a homogeneous body, possessing a given magnetizing function F (OÏL), magnetizes uniformly in a given field, so that its magnetization has the intensity M at each point, a body of the same shape, placed in the same field and having a constant magnetization coefficient A , given through equality A- = F(M), will be magnetized in the same way. To this theorem corresponds a reciprocal, very important for the object of our research. This reciprocal, which can be proved like the previous theorem, is stated as follows: Let us imagine a magnetic field and a magnetic body which, in this field, magnetizes uniformly, whatever the value k of the constant magnetization coefficient that we attribute to it. The intensity of magnetization dV^ q a' it then takes is related to k by the relation (I) d^^^ik). If we attribute to this body a magnetizing function, which 200 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. cone F (OÏL), variable with V magnetization intensity, it still magnetizes uniformly, and its magnetization is the same as if it had a constant magnetization coefficient defined by V equality (2) F[|(/0]-A-o. A sphere, an ellipsoid, placed in a uniform magnetic field, magnetize uniformly whatever their magnetization coefficient. The previous theorem can therefore be applied to these bodies. Let us propose, for example, to find the magnetization taken by an ellipsoid whose magnetizing function is F(Dl'L) in a uniform magnetic field, whose potential function is F, G, H, K being four constants. If the ellipsoid has a magnetization coefficient A", the components of the magnetization are given by the equalities [Book VIII, Chap. II, equalities (i)]. ■2 A IX 7-"' "k^ [X, V being three constants that depend on the shape of the ellipsoid. We deduce from this I F2 G2 H2 1 01^2 = .1,2+ D!,2+ 32 = /,2 _^ + r , l ^i - L(h-2A:X)^ {i-h-2kii)^ (i-l-2A-v)2J The function <^ (k) is thus defined here by the equality r F2 G2 H2 y (4) ^(/c) = /, [^-7^-^)^ + (i+/.A-[.)^ ^ (1+ 2X-v)2j ' and we have "^~ i-f-2XF(orc) ' (5) \ ^''>- - ^ttWtG, ^ ' I i -t- 2 [a F (OÏL) \" I + 2V F(OIL) ' CHAP. IV. - THEORY ON MAGNETIC BODY. aoi with (Sbis) DTL ^ F(;)rt) } î7ip^p|-^)p - h^.^^F(OIL.)p ^ L' + '^ vT(;;iL)J^ f " Equations (5) and (5 bis) will determine the magnetization of the ellipsoid when we know the form of the function F(DÏL). Indeed, the equality (5 bis)^ solved with respect to OÏL, will let us know as a function of F, G, H the value of the magnetization intensity taken by the ellipsoid. Once this value of OÏL is known, we will calculate the corresponding value of F(OIL), and the equalities (5) will then show the components of the magnetization at each point. The volume of the ellipsoid will be -abc, <7, b, c being the three half axes. The components of the magnetic moment will then have the value . / A 4 , F (.OÏL) " A =: - TT abc \ ■-, , .,,■■ , F, l J I -t- 2 X 1^ ( OIL ) ^A^ R 4 , F(OIL) (t") B - ;r 11 abc / G, \ 3 1+ :2fJL F(OIL) G = - T^abc " . H. At a point outside the ellipsoid, the magnetic potential function of this body will have the expression [Book VIII, Chap. Il, equality (9)], 2F(DV^)LFx 2F(01L)MGjK 2F(01L)NH2 (7) X)ia;,y,z)=. i-t-'2XF(01L) r + 2iJLF(0IL) i-h2vF(0IL) L, M, N being three functions of x, y, z. For a sphere of rajon R, we have a = b = c = H >. = F = V --= i 3 TC 2 3 ttRs r. - \i - N - (^2_^^2 202 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. The previous equalities thus become (8) <Âa = ■Dl) Fcm) F (OÏL) H-|7rF(01L) F (.m) i + |-TrF(Ort) F, G, H; (8 bis) DrL = F (011) (F^N-G2^H2)2; i-i-^7rF(01L) (9) i7rF(0lL) R3F, i + |7:F(01L) B = 7rF(01L) K^G, i-f-iTTF(01LO i7rF(0lL) G = ^- R3H; H-^7rF(DlL) (10) t){x,y,z) TtF(DlL) R5 I+^7rF(01L) (^■2-+-j2^z2)2 -(F^+ Gj'-i-H2). This last equality allows us to show that the method indicated in Book VIII, Chapter II, § 4, for determining the magnetic inclination by the deviations that a sphere of soft iron undergoes with a declination needle, remains legitimate, even if we admit that the soft iron, instead of presenting a constant magnetization coefficient, presents a magnetizing function that varies with the intensity of magnetization. The only requirement for this method to give accurate results is that the sphere be homogeneous and, a condition more difficult to achieve, that the iron which constitutes it be perfectly soft. CHAP. IV. - THEORIES ON MAGNETIC BODY. 203 This formula allows us, by the method indicated in Book VllI, (Chapter II, § 3, to determine the particular value that F(OIL) takes for a sphere of soft iron placed in the terrestrial magnetic field; this value once known, equality (8 bis). provides the corresponding value of the variable 011. § 3 - Determination of the magnetizing function. - Saturation. The previous method makes known only a particular value of the function F(OIL), whereas it would be of great interest to know the values of this function for a large number of values of the variable. This can be achieved in the following way: An ellipsoid is subjected to the action of a uniform field, of variable intensity F, directed along its long axis. This ellipsoid takes on a magnetic moment, also directed along its long axis, and whose magnitude is determined by the first of the equalities (9). The measurement of this magnetic moment gives F (OÏL); the corresponding value of Oll/ is given by the equality (8 bis). Measurements of this kind have been made by W. Weber (*). A cylindrical bar, which can be likened to a very elongated ellipsoid, is placed inside a spiral through which a current flows and which generates a uniform field (-); the apparatus acts on a magnetometer subjected to the action of a second spiral which compensates the effect of the first. From the experiments of W. Weber, G. Kirchhoff (^) deduced the values of F(01l/) as a function of 011/, or rather the values of )v(n'C) as a function of (11'^)-. Here are the results of this calculation, in C.G.S. units, i (') w. Weher, Elektrodynamisc/ie Maasbestimmungen, p. 629. (") See Book XV, Chapter VII, § 1. (') G. KincHHOFF, Ueber den inducirleii Magnetismus eines unbegrenzten Cylinders von weichem Eisen {Journal de Crelle, Bd. XLVIII, p. 348; i854. - KirchhoJjTs Abhandlungen, p. 221). 204 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. u. >.("). 3oi -23,5 823 - 13,5 1184 - 10,2 l5l2 - 8,4 1773 - 7,4 2080 - 6,4 2397 - 5,7 8. 9. 10. 11. 12. 13. 14. u. Mu). 2484 - 5,6 1975 -6,7 i583 -8,1 1297 - 9,5 967 - 12,0 612 - iG,9 296 ->.5,o For the earth's magnetic field, we would have u = I .78. We can see, by the experimental results we have just reported, that )^(n\'^) decreases rapidly in absolute value when W'Ç grows beyond all limits. According to G. KirchhofF, we can write the function fijl"^^) tending to a finite and positive limit P when W'Ç grows beyond any limit. If this is so, the following proposition can be proved: If a soft iron mass of any shape is placed in a magnetic field whose intensity increases beyond any limit, the intensity of this magnetization tends towards a limit P which depends exclusively on the nature of the soft iron; moreover, the limit direction of the magnetization coincides with the direction of the field. Under these conditions, the soft iron mass is said to be saturated. The equations of the magnetic equilibrium are, indeed, Dl) dx e = X(nt,,)^|. Let X3 be the magnetic potential function of the soft iron mass and '^ the magnetic potential function of the field. According to CIIAP. IV. - THEOREMS ON MAGNETIC BODIES. the equality (it), the previous equalities can be written tX, = - p{ T dx (upy dx (i-i) llï, = = -p{ L(n'>)^ p{nv)\ T ^7- IV Let's put (i3) ï = âx (nxg))2 dz T - ,, . , ^-^ cm asÇ) r)^t?; Let us assume that at least one of the quantities --5 -r- , -r;- grows ^ >■ 1 dx oy dz beyond any limit, but that each of the three quantities a, |S, Y tends towards a finite limit. The intensity of the field will grow beyond any limit, but its limiting direction will be determined^ its directing cosines will be the limits ", 6, c of the quantities a, P, y defined by the equalities (i3). It is easy to see that, if we assume infinity of at least one of the > -- > the equalities (12) will be verified by dx dy dz A. = P: quantities posing (l4) jlli, =:PP, which will demonstrate the stated proposition. Indeed, ,^1,, li'o, 3 tending, according to these equalities (i4)j towards ,. . P . ., 1 A I dV) dXD dXn . ' . . rr , finite limits, it will be the same for -r- > -r- ^ -- : the quantity IlV ' dx df dz ^ will grow beyond any limit, and its ratio to II will tend to unity; in other words p(\lV) will tend to P. The equalities (12) will thus be verified. 206 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. This theorem, which, like all those developed in this Chapter, is due to G. Kirchhof, explains the phenomenon of saturation which we encountered at the beginning of this Book IX as a serious objection to Poisson theory. KirchhofF, explains the phenomenon of saturation which we encountered at the beginning of this Book IX as a serious objection to Poisson's theory. We have just seen that, for large values of (IIQ)-, the P function ).(nO) for soft iron tends to o as (nû)2 for small values of (ITO)-, this quantity has, on the contrary, an increasing absolute value. Lord Rajleigh (' ) found that, for At these low values of (IIO)-, the function }.(nQ), relative to iin annealed Swedish iron, could be represented by an expression of the form ).('nQ) = --[6,4 -^ '5,i2(nî2)^J. Mr. Paul Janet (-) found, by a different method, for another soft iron, X(nQ) = -[6.3 + 3.9([mr-]. These results show that the function F (011) first grows with OR, when DIL starts from o, passes through a maximum, and then decreases and tends to o as -r^r when DIL tends to B. Recently, Dr. E. Beltrami (■') proposed, for the À (nQ) function related to large values of IIQ, the following form: PK v/p-^-t-K2.rii2 p being the maximum magnetization intensity and K a constant. This formula, according to the calculations of Mr. P. Pizetti, represents very exactly the results of W. Weber's experiments, if one does P = i3i35 .2 ± 2o3.9, K = 29.0369 ± 0.55. (' ) Lord Rayleigh, Philos. Magazine, t. XXIIf, p. 225; 1887. (' ) Paul Janet, Étude théorique et expérimentale sur l'aimantation transversale des conducteurs magnétiques, p. 83. (') EuGENio Beltrami, Considerazioni sulla teoria matematica del magnetisme {Memorie delia Reale Accademia délie Scienze dell' Istituto di Bologna, série V, t. I; 1891). CHAP. V. EQUILIBRIUM AND MOVEMENT OF MAGNETS. 207 CHAPTER V. EQUILIBRIUM AND MOTION OF A MAGNETIC MASS IN THE PRESENCE OF PERMANENT MAGNETS. ^ 1 . - General equations of motion of a perfect body soft. Let us consider a system iorinated by permanent magnets, which we will designate by the index 1, and by a perfectly soft body, which we will designate by the index 2. Let us propose to determine the laws of motion of body 2, or, in other words, to find the forces that act on body 2. This body 2 can be any: solid or fluid, compressible or not; it is only subject to carry at each instant the magnetic distribution that is appropriate for equilibrium; so that at each instant, we must have, at any point {x^ty^-, ^2) of body 2, (') 1 - -F(Dll2,a,p, . 1 -F(31U,a,?, . {-> -F(Dll2,a, ^, . "Ç being the magnetic potential function of the whole system. This being the case, let's look for the external forces that must be applied to body 2 to keep it in equilibrium; the forces that act on body 2 will be the forces capable of balancing those. If we know how to determine the first ones, we will know by that very fact the last ones. Let 3" be the internal thermodynamic potential of the system; let 0^ be the variation that it experiences when the various 2o8 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. points of the body 2 a small displacement, accompanied by small variations of magnetization such that the equations (i) remain verified. Let ûS be the work done, in the considered displacement, by the external forces applied to body 2. If these forces maintain body 2 in equilibrium, the infinitely small modification that we have just considered will be reversible, and we will have 5G Ort = G. The external forces that must be applied to body 2 to keep it in equilibrium perform, in any virtual displacement of the perfectly smooth body, work equal to the total variation of cF. These forces are intended to balance the forces acting on body 2. The forces acting on the perfectly soft body 2 have therefore for potential the internal thermodynamic potential of the system. Among these forces, those that can be considered as magnetic in origin will have as potential the quantity (2) ^ -H r#i ( OK 1 ) dVy 4- l'.f 2 ( Dli 2 ) dv^. Since the magnetization of the permanent magnets does not vary, the amount J' remains invariable during the movements that can be imposed on body 2. Let t)( be the magnetic potential function of body 1 and ï')2 the magnetic potential function of body 2. We will have The term Al cl,2 dx. X. i u\ dv9. does not vary in the displacements and deformations imposed on body 2. If we then notice that we can subtract from the potential of the forces acting on the body 2 the terms that the displacements CIIAP. V. - lÎQUILIBRE AND MOVEMENT OF MAGNETS. 209 If we consider that the deformations of the body 2 remain invariant, we can see that we can replace the expression (2) of this potential by the expression (3 > '--It .1,. dvi 'J\ .l>2 , dV, 0X9 dvi -J^2{1 d'(i',)dVi. In the case where body 2 magnetizes according to Poisson's theory, we have, by designating by k-i its magnetization coefficient [Chap. II, equality (27)], 2 AO and the previous equality becomes (') ." r=/| -Uo Oxi dv^ If 1 toV, -VS2 ^ - 0X9 dv-i -:/ u^t - dv-i. This very expression can be transformed. We have, indeed, Moreover, if body 2 is magnetized according to Poisson's theory, the equalities (i) become s^ia-T - '12 \l,2 - - A2 W2 - - A 2 dxi - ^/2 ' So we have c^.r2 dvi. dVi, -y ''^2 J W and equality (4) becomes Now, if body 2 were a rigid body and if the magnetization it carries (Hait assumed permanent, the mutual actions of the two magnets (') GoTTLiEB Adler, Ucber cUe Energie inagnetiscli polurisirter Korper,nebst Anwendung der beziiglichen Formeln au/ Quincke's Méthode zur Destimtnung der Magnetisirungszahl {Sitzungsber. der Akad. der ff issenscha/te.i zu Wien. XCII, 2' Abth; i885. - Wiedetnann's Annalen. Bd. XXVIII, p. Socj; i8s6). D. - II. i4 210 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. 1 and 2 would have as potential [Book YII, Chap. I, equality (lo)], the quantity (6) (^=f\j[,,^ The comparison of equalities (5) and (6) dvi (7) $ = 2é?. When a perfectly soft body, placed in the presence of a permanent magnet, becomes magnetized in accordance with Poisson's theory, the potential of the magnetic actions that it undergoes at a given instant is half the potential of the mutual actions that would be exerted at that instant between the two bodies, if the soft body were, at that instant, transformed into a permanent and indeformable magnet. This proposition is due to M. J. Stefan (' ). It is correct only for bodies that magnetize according to Poisson's theory. Let us return to the c^s where body 2 is any perfectly soft body; equality (3) can be written '-Rh ^(r),-4-t%) tox^i. ii{3\i'i)\ dv-i- - mU=>2 or, by virtue of equalities (i), dvt. di>i, According to a known transformation [Book VII, Chap. III, equality (17 bis)], we will have (9) ij II dX'i 8-!zJ the integration that appears in the second member extending to the whole space. ( - ) J. Stefan, Ueber die Gesetze der elektrodynamischen Induction {Silzungsberichte der Akademie der Wissenschaften zu Wien, LXIV, 2° Abth., p. 19.3 ; 1871). CHAI'. V. - EQUILIBRIUM AND MOTION OF MAGNETS. 211 Let's put "iTi and equality (8) will become (11) r=- -^- fuXD^di-^ fWi{D\L2)di^2. The function W2(^1^2)7 defined by equality (lo), will often be represented in our calculations; let us briefly indicate its properties. If we remember that we have [Chap. II, equality (5)], we see that we can still write or by designating by pi2 a certain value of 011-2 between o and OR 2, i" Let us first consider diamagnetic bodies. For these bodies, the function F2(011'2) is negative. If its absolute value is independent of OlLo, decreases as OIL2 increases, or increases weakly with OTL2, we will be assured that the absolute value of F2(0n.2) is smaller than the absolute value of 2F2(|i.2). We will thus have (i4) ^2(an.2) or decreasing when SÏL^ increases, the function ^r2(01L2) is negative. 2° Let us consider secondly the magnetic bodies. For these bodies, the function F2(0Tl2) is positive. It is then easy to see that, for a magnetic body whose magnetizing function is independent of D]1.2, slightly increasing with D\Lo, 212 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. OR decreasing when 0112 grows, the function W2{3\i2) is positive: (l5) T2(01i2)>O. We can admit the following hypothesis, which we will use frequently: For all magnetic bodies, the function ^F(,')ll) is positive. In the case where the Poisson approximation can be accepted, the magnetizing function F2(<01L'2) can be replaced by a magnetization coefficient k^^ independent of DlLo- Equality (3) then gives (i6) XF,(01V2) = ^^ Comparing this equality with equality (27) of Chapter II, we see that we have, in this case, (17) ^^{d\l^) = fu_{d\L,). With the information we have just obtained about the function ^'*2(5l^2)> the equality (i i) gives us the following proposition: The function j" is always negative for any magnetic (but not diamagnetic) body, placed at a finite distance from any permanent magnets. If the perfectly soft magnetic body 2 is located at an infinitely large distance from the permanent magnets 1, it is no longer magnetized, and the second member of equality (11) is equal to o. The quantity J" thus takes its greatest value when the perfectly soft magnetic body is infinitely far from any permanent magnet. If we now observe that, according to the properties of the internal thermodynamic potential, the magnetic forces acting on the soft body always tend to move it in such a way that the quantity J" passes from a greater to a lesser value, we see that we arrive at the following law: A perfectly soft magnetic body, placed at a distance of Cn.VP. V. - EQUILIBRIUM AND MOTION OF MAGNETS. 21 3 In the case of a very large number of permanent magnets, they always tend to approach the magnets. This proposal justifies the name of substances attractable to the magnet, often given to the magnetic subslances pai'fa'soft. Equation (i6) shows that, in the case where the magnetic body magnetizes according to Poisson's theory, equality (i i) can be written (.8, r.-fJno,*-.;/i|l*,. This form has often been used by authors who have dealt with magnetism. Let us return to equality (2) which gives, in the most general case, the expression of the potential §' of the magnetic actions which act on the perfectly soft body 2. When the perfectly soft body 2 is given an infinitesimally small displacement compatible with the bonds to which it is subject, the components X^^ DÎjo, ©2 of the magnetization at each of its points undergo variations ^X^, Bi)î)25 ^^2, so that equalities (i) continue to be satisfied. These variations in the position, forijie and magnetization of body 2 result in a total variation 8j' of the function 3' . This variation can be seen as the sum of two others. Let us imagine that we impart to body 2 the displacement and deformation that we want to study, but that each point, while moving, causes its magnetization to change neither in magnitude nor in orientation with respect to the coordinate axes. The quantity §' would undergo a variation 8, §' . On the other hand, let us suppose that, leaving body 2 immobile, we vary the components X^, i)î>25 ©2 of the magnetization at each point by 8cAo2, 8i)l)2, 8^2- The quantity §' would undergo a variation 82^'. We will obviously have (19) Zt=^J'-^^J'. But we have, by reproducing the reasonings exposed in Chapter II, 21 4 BOOK IX. - MAGNETIZATION AND THERMODVNAMICS. OR, according to the equalities (i), Zii'= o. Equality (8) thus becomes (20) 8.f'=â,.f'. When a perfectly soft body deforms and moves in a magnetic field, the magnetic forces applied to it do work equal to the sign of the variation that the quantity ^' would undergo if each particle of the body 2 were to drive its magnetization without changing either its size or its direction. Instead of assuming that, in the first modification, the magnetization of each point is driven by keeping an invariant magnitude and an invariant direction in space, one could assume that the magnetization at each point is driven by keeping an invariant magnitude and an invariant orientation with respect to three axes related to the material particle of which that point is a part. The previous proposition would remain true. The new form that the previous proposition takes is easily applied to the case where body 2 is an undeformable solid. The displacement of this body results from a translation whose components parallel to the axes are ox, ojk, 05, and from three rotations ùk, B^u, Sv, around the three coordinate axes. Therefore, we can easily deduce from the previous proposition the equality 8^' = (21) 8x 1 ( wlas -f- \l'02 dx^ây^ - I- C*^2 àx^ ai he dz2 âx-i - (^K2 toXi + 1)1)2 -4-1)^5 '■ àyl This expression of oJ' gives back, for expression of the magnetic actions which act on the body 2, the expressions given by the equalities (12) of the Book VIT, Chapter L CHAP. V. - EQUILIBRIUM AND MOTION OF MAGNETS. 21 5 § 2 - Instability of the equilibrium of a magnetic body in the presence of permanent magnets. The eagle (21) will allow us to answer in a completely general way a question raised by the researches of Sir W. Thomson (' ) and solved by him for a very small and not very magnetic body. This question is the following: An undeformable magnetic mass devoid of coercive force, placed in the presence of permanent magnets, is subject to external forces which reduce : 1° At a normal and uniform pressure at any point of its surface; 2° A force constant in size and direction applied to each of its elements. This is essentially the case for a mass placed in the air and subjected to the action of gravity. This mass takes an equilibrium position determined by the considerations developed in the previous paragraph. Is this equilibrium position stable? The external forces admit a potential W. The system then admits a total thermodynamic potential cp which is the sum of the internal thermodynamic potential § and the potential W of the external forces o = .f -t- W. If, for all the virtual modifications that leave the form and the physical and chemical state of the various parts of the system unchanged, © undergoes a positive variation, the system is in a state of stable equilibrium. The first variation of cp is identically null, since we suppose realized the conditions of equilibrium studied in the preceding paragraph, which, all, derive from the equality 0-)' Oto = O or 8i+oW = o. (' ) Sir W. Thomson, Remarks on the forces experienced by inductively magnetized ferromagnetic or diamagnetic non-crystalline substances ( Philosophical Magazine, t. XXXVII, p. 241; iS5o. - Beprint 0/ papers on electrostatics and magnelism, n' edition, p. 5i4). 2l6 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. We are therefore led to investigate whether the second variation of 'f is positive. We will start by looking for whether 0-0 is positive, not for any virtual displacement of the body without coercive force, but only for any virtual translation; in these conditions, it is not difficult to see that and that the question comes down to whether 0- j" is always positive. If the only forces inside the system are the magnetic forces, we will have (22) ^Ki = ^^^'. According to equality (21), we have ' dx\ âX2 ày^ ' dx^ d dx^ ôy^ ôyi (^^2 toz^ ) Let us imagine that Ton imparts the translation (0^, qy, §5) to body 2 a second time. Let us denote by 0' J-o, o'oIÎjo, o'S2 the variations that, according to the equalities (i), a similar translation causes X2, i)l)2, ©2" to experience, observing that these variations are functions of 8^, Sj^, ùz and must not be confused with the arbitrary variations that we will represent by ô^l>2, Si)l)25 '^^iThe preceding equality will give us ^y ^z I Xi ^ ^ h 1)J>2 , " , -1- G2 -; e-s- 1 dvi (23); 8./( ^^J\dx,dy,-' ' dyl - -- ■ dy.dz, J \dx2dz-î oyidz^ dzl ) --ri 0' X^ 4- ;, -- r^ ô' iJla + -. T^ Ô Gj ) ^1^2 Oa^l 0372 0/2 00^2 ''-32 tf 0.^2+ -^-5- oalli2+-^ T- ûG2 \dvi CHAP. V. - EQUILIBRIUM AND MOTION OF MAGNETS. 217 This expression of 8^J' can be transformed. -l.o, iil)2? ^2 must, at each instant, verify the equalities (i); this condition can still, if we refer to the method followed to establish the equalities (i), be stated in the following way: We have at all times, whatever ô^lo, ôilbo? ^So" / Ox-i 0x9 Ù-X>9 dv" =: O. This condition must be verified when I body 2 occupies its initial position with its initial magnetization; it must still be verified after body 2 has been subjected to the translation (5.C, Sj', Ss) and to the magnetization (^Xi, 1)1)0, Sa) the variation (3'al,2, 8'\li)2, ô'So)' The first member of this equality thus experiences, when one imposes this translation to the body 2 and this variation to the magnetization that it carries, a variation which must be equal to o, whatever 8.A>2, oïl^a, SSo We must therefore have, whatever ùâo^i Si)l)2, 080, -~ 8 1)1)2 + ^ r^ 8S9 1 dv^ C)X2 (^^2 y (Jv92 2 - A^ "^^'i.-^^'^ (5jl,2 8'cil>2+ SH''>2 8'DÎ'2-(- 8328'S2)^"'2 ^ 1/1^2 "1/1^2 This equality, having to take place whatever So,l,2i Sil^a, 882, will still take place if we pose 8JI02 = û'cilo2, 8\)b2 = 8''l)l)2, 882= 8' S2. The form that it takes by this substitution, compared to the equal (■>.5) -218 BOOK IX. lity (23), gives MAGNETIZATION AND THERMODYNAMICS. 82 -F' z /(^ '^''' dj.2 -i oy oz dx\ ôy^ (J.Z2 dxl ÔZ-2 âxi oy^ aPo2 a)î.2 \\\ ^72 c*^! àyi àz^ ' C'*"2 't>^2 <^-32 dx'^ dzi '- dyl ôZi ''■ dy^ dzl ■ dx.^ dy dvi (^) dv-2 (3) dV2 (i) dv.2 (5.) dv dv" 112 <:/tJlt2 [( 5M>2 )2 + ( a' 1)1,2 )- + ( 0' S2 )2 ] C/P2 J OIL2 25,082 is given twice in a row to the magnetization at a point of body 2, and this is true whatever oJl92, ôi)l)2, 532 are, provided they are not identically zero. This will happen in particular if one makes unless ô'^Ag^, 0' \\,2i o'So are identically zero. The quantity o.'f ^' is therefore positive, unless we have, at any point of the body 2, 0''Â>2=O, 0'l)i)2=O, 0'32=O. According to equality (24), these last equalities cannot take place, unless one has, at any point of the body 2, d^Vi (JHOi d^Vi -^ - r = O, - = O, - - 7- = o, i.e. unless the field in which the body 2 is placed is uniform. In this case, the two quantities SJ J', ù'^ê' are 220 BOOK IX. - THERMODYNAMIC KT MAGNETIZATION. identically zero, and the same is true for variations of all orders of ^' . We thus arrive at the following conclusions: 1° If the magnetic field in which the body is placed is uniform, the balance of the body is indifferent. 2" If the magnetic field in which the body is placed is not uniform and if V magnetization taken by this body is stable for the position it occupies, there are certainly translations for which 0^ ^ is negative, and V balance of the body is unstable. We have seen that, on all known magnetic bodies, placed in a determined position, the magnetization takes a stable distribution; the equilibrium of a magnetic body, indifferent in a uniform field, is unstable in a non-uniform field. CHAP. VI. - IMPOSSIBILITY OF DI.VMVGNETISM, CHAPTER VI. IMPOSSIBILITY OF DIAMAGNETIC BODIES. We have left aside, in the three preceding Chapters, the dlaniagnetic bodies, i.e. the bodies for which the function F(OIL) would be negative; let us now return to the study of these bodies. A perfectly soft, immobile body being placed in the presence of permanent magnets, the second variation of the internal thermodynamic potential of the system has the expression [Chap. III, equality (i6)J, oKj = - 1 IIQ of HtcJ ^y^àF(:)]1,a,^,...) the various letters contained in this formula keeping the meaning given to them in Chapter III. Suppose that the body is a diamagnetic body; its magnetizing function F(.')IL) is then a negative quantity. Suppose, moreover, that this coefficient has a very small absolute value. The equations of magnetic equilibrium dx G^-F(,m,a,,3,.,.) A X^-F{0\i,oi,^, Dl, = -F(Oll, a, 3, ...) 222 BOOK IX. - AIMANTATION AND TIIERMODYNAMIQLE. show that the value of DM. is a very small quantity of the same order as F(01l, a, ^, . . .). The quantities ~ fnadu, 4- f^^dv OTT y oit J are finite quantities. The quantity -J F(OIL,a,p, ...)^'^ is a negative quantity whose absolute value, very large, is of the order of " , ,. ,^ ^ - If the function F(orL) varies little, so that the quantity d ¥ (DXl) . |. 111 I - , T-. / NI is negligible in front of the quantity r (OR), the quantity r ()F(,m,a, 3. ■■■) J = ~ / , ,-, , ..., r, TV-"^ dv J [l^(Oll,a, 3, ...)J^ will be negligible in front of the quantity J. If - -^ - is not negligible in front of F(Oli), it will not be the same. If the absolute value of F(D1L) decreases as OÏL increases, J' is very large and negative. J' is very large and positive if the absolute value of F(OllL^) increases with DU. Thus, for a diamagnetic body whose magnetizing function has an absolute value that is always very small, that remains constant, that grows very weakly or that decreases when V magnetization increases, ù-^ is always negative. ^ can admit a maximum and only one, but cannot admit a minimum. Without having to worry about the direction of the variation of F(Oli), we can arrive at an interesting conclusion, although less complete than the previous one. The magnetization of a perfectly soft body can always be varied in such a way that at any point m - o. If we remember, in effel, that m is defined by the equality oOlL = m ot. CHAP. VI. - IMPOSSIBILITY OF DIAMAGNETISM. 223 we can see that it will be sufficient, in order to reach the goal we have just indicated, to vary the orientation of the magnetalion at any point without varying its intensity. For a similar variation, we have J'^o. We can therefore state the following proposition: A diamagnetic body, whose magnetizing function has a very small absolute value, being taken in any state of magnetization, one can always impose on this magnetization a variation such that the quantity 8-ef is negative. 3' cannot therefore have a minimum for such a body. The propositions we have just demonstrated lead to the following conclusion: -5"///* a diamagnetic body whose magnetizing function is always very small, there can be no magnetic distribution corresponding to a stable equilibrium. This conclusion must still remain true for a diamagnetic body whose magnetizing function is not very small; this is what we will demonstrate, using the properties of the ^^'(Dri) function, established in § 1 of Chapter V. Let us consider a system formed by permanent magnets, which we will designate by the index 1 and by a body devoid of coercive force, which we will designate by the index 2. The internal thermodynamic potential of this system can be written, denoting by ^^^ the magnetic potential function of the permanent magnets 1 at a point of these magnets; by XJP2 ^^ the potential function of these magnets at a point of the body 2; by X)^ the potential function of the body 2 at a point of this body, and assuming the system not electrified to'^Ai , riK to^ ">■.) It dxi II J cl,. dxi dvi "i- J II ^^2 II c/ J This expression is general; it is accurate, in particular, if the 224 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. The magnetic distribution on body 2 is an equilibrium distribution. Let us consider the diamagnetic body and the magnets in a certain position, and suppose that a certain stable equilibrium distribution corresponding to a minimal value J'o of the internal ihermodj'namic potential has been found on the diamagnetic body; then, let us suppose that, the bodies being placed in the same position, all the particles of mass 2 are given the same magnetization, except for the particle dx^ dy^ dz^ = dv^-, which we will assume is not magnetized. The internal thermodynamic potential of the system will then take a new value ,?,, and we have dv-i-^ .'fi{Ù\\,_)d\>z, rfa-^ the quantities which appear in the second member having all the values which they have in the state of equilibrium considered. Now, in this equilibrium state, we have atXa o\.2 = - i%(orL2) Dî,, F^c-mo -F.iOTL,) From these equalities we deduce ;)ii? ^''i{0\U) and, therefore, [-■ j, = ,T2(3rt5 nrL2 dVr, F, (OR 2). or, according to equality (i) of the previous Gli Chapter, .fo- J,=.-VF2(,m2)^i^2. If body 2 is a diamagnetic body whose magnetizing function is independent of 011, increases weakly in absolute value with 011, or decreases in absolute value when OÏL increases, we have CHAP. VI. - IMPOSSIBILITY OF DIAMAGNETISM. 225 according to the inequality (4) of the previous chapter, ■^0- ■-'?i> o. We see, from this, that if, for one of the diamagnetic bodies verifying the previous restrictions, we considered a magnetic equilibrium distribution corresponding to a minimum of the thermodynamic potential, we could always find a distribution in which the thermodynamic potential would have a lesser value than in the considered state. It is, according to this, very likely that, for such bodies, there is no minimum of the internal thermodynamic potential, a proposition that we know to be true when the magnetizing function is very small. But here is an entirely general demonstration of the fact that, on a system of which any portion is diamagnetic, the thermodynamic potential can never present a minimum. Let dv be a volume element taken in a diamagnetic region. Let us rotate the magnetization of this volume element, so that its components JU, ilî), S vary by Sol>, oiJl), o3, without the magnetization Oit changing magnitude. We will have ( i ) X S^.b -T- \)l) 01)1) -)- 3 58 = o and o.j" = -- o-.Id + ,- oi)b + -- o3 dv. ^ ox ay oz From this last equality, we deduce \ox oy oz I OR, under the conditions of equilibrium, ox C =-F(01L)^, D. - II. 226 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. But equality (i) gives X B^X -h M\> o2Db + G 82S + (0.1,)'--+- (oi)'o)24- {oBy-=o and we have, therefore, s^ ^ = p^ [( 5.1, y -H ( 8Di y + ( sa)2 ] ^^,. If F(Ol'L) is negative, so is 8-rf. Thus, 5/, in a system, a region, however small, is diamagnetic, there can be no stable magnetic equilibrium on this system. We can therefore state, without rectriction, the following proposition: The principles of thermodynamics do not allow the existence of diamagnetic bodies, i.e. bodies whose magnetizing function is negative. Let us briefly indicate here the research that led physicists to this proposal. As early as 1887, we gave (*) the expression of S^^, but without noticing that this quantity must be negative for diamagnetic bodies. Later (2), we noticed that, if one assumed equilibrium to be established on a system containing diamagnetic bodies, one could always define a state of the same system corresponding to a lower value of the internal thermodynamic potential. But, instead of concluding that stable magnetization on diamagnetic bodies was impossible, we simply concluded that there could not be a unique state of equilibrium on these bodies, a conclusion which seemed to us to be in conformity with certain experiments of M. Joubin, which will be discussed in Chapter VIII, § 3. It is this conclusion that we set forth in our new Theory of magnetization by influence, based on Thermodynamics (' ). Shortly thereafter, Mr. Parker ('-) acknowledged that there was a contradiction (') P. DuHEM, Sur l'aimantation par influence {Comptes rendus, t. CV, p. 798; 3i octobre 1887). {") P. DuHEM, Sur l'aimantation des corps diamagnétiques {Comptes rendus. t. CVI, p. 786; March 12, 1888). (' ) Annales de la Faculté des Sciences de Toulouse, t. II; 1888. {*) John Parkkr, On diamagnetism and concentration of energy {Philosophical Magazine, S" series, t. XXVII, p. 4o3; May 1889). CUAP. VI. - IMPOSSIBILITY OF DIAMAGNETISM. 227 between the existence of diamagnetic bodies and the Carnot-Clausius principle; but, instead of concluding that diamagnetic bodies are impossible, M. Parker proposed to modify the Carnot-Clausius principle. It is then that we developed (' ) the considerations that we have just read. . At the time we published these considerations, we were not aware of a remarkable Note (2) that Mr. E. Beltrami had published shortly before Mr. Parker's Memorandum. In this Note, Dr. E. Beltrami demonstrates that, for a fully diamagnetic system, the quantity which represents, by adopting the Poisson approximation, the magnetic part of the internal thermodynamic potential (M. E. Beltrami says the energy) is negative. This result," adds Mr. E. Beltrami, "leads to another one, which is no less improbable. It is known that if, to the magnetic distribution induced in a body by external magnetic actions, given and invariable, one superimposes another magnetic distribution, the potential of the whole system increases by a quantum which is simply equal to the potential of the distribution superimposed to the induced distribution. It follows from this, taking into account the previous result, that if the induced body is paramagnetic, the potential increases when the induction equilibrium ceases, but on the contrary, if the body is diamagnetic, the potential decreases. In the first case, therefore, the total potential would be minimum in the state of equilibrium; in the second, on the contrary, it would be maximum, so that the equilibrium of magnetic induction would be unstable. " (') P. DuHEM, Sur l'impossibilité des corps diamagnétiques {Comptes rendus, t. CVIII, p. 1042; 20 mai 1889). - Des corps diamagnétiques {Travaux et Mémoires des Facultés de Lille, Mémoire n° 2; 18S9). (') E. Beltrami, Note fisico-matematiche, lettera al prof. Ernesto Cesàro {Rendiconti del Circolo matematico di Palermo, t. III; session of March 10, 1889). 228 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. CHAPTER YII. MAGNETIZATION OF A MAGNETIC BODY IN A MAGNETIC MEDIUM. § 1 - History. Thermodynamics leads to the rejection of the existence of diamagnetic bodies, as does the theory of magnetization by influence devised by Poisson. But, whereas the incompatibility of diamagnetic bodies with Poisson's theory had simply led most physicists to reject the hypotheses on which Poisson's theory rests, their incompatibility with the principles of Thermodynamics must necessarily lead to the rejection of their existence, unless one wants to reject the axioms of Clausius and Thomson, that is to say, to admit the possibility of creating, with diamagnetic bodies, instruments capable of producing a perpetual motion. Therefore, there can be no diamagnetic bodies as such, i.e. bodies whose magnetizing function is negative. However, nature presents us with bodies, such as bismuth, whose magnetizing function seems to be negative. How should we interpret the existence of such bodies? The answer to this question seems to have been given by Edmond Becquerel (' ). According to Edmond Becquerel, all bodies are magnetic; but they are all immersed in an ethereal medium which is also magnetic; they then seem to be magnetic or diamagnetic. (') Edmond Becquerel^ De l'action du magnétisme sur tous les corps (Comptes rendus, séance du 21 mai 1849. - Annales de Chimie et de Physique, 3" série, t. XXVIII, p. 288; i85o). CHAP. VII. - AIMANTATION WITHIN A MEDIUM. 229 They are in fact more or less magnetic than the medium in which they are immersed. Here is how M. Edmond Becquerel states this hypothesis: "A body placed at a distance from a magnetic center is attracted to this center with a force equal to the difference between the specific magnetism of this body and that of the environment in which it is immersed. Or, in other words, the action of magnetism on a body is the difference of the actions exerted on this body and on the displaced environment. " Mr. Becquerel adds: "We can account for this principle by means of a demonstration similar to the one used to prove Archimedes' principle. "Let A be a magnetic center placed in the middle of a space filled with a fluid that can be alloyed to the magnet. In this position, there will be a certain state of equilibrium, according to which each point of this fluid will be at rest, and there will only be an increase in pressure as we approach A, pressure exerted by the medium that serves to transmit the magnetic actions. According to this, if we consider an isolated mass M of this fluid, this mass will be attracted towards A with a force that can be represented by y"; now, since there is equilibrium at all points of the medium when the fluid surrounds A, it is therefore necessary that the action of the magnetic center A on the surrounding fluid M gives a resultant equal to y and directed in the opposite direction: this is equivalent to a repulsion equal to - f. Let us now suppose that another substance of the same volume is substituted for the mass M; the attractive force from M to A will be greater or smaller qne/, depending on whether this substance is more or less magnetic than the medium. Let us represent this force by F; the force by virtue of which the mass will move towards the center A will thus be (F - /), the repulsion - / existing in this case as well as previously, n PliickerC), studying at the same time as Edmond Becque (') J. Pllcker, Ueber den Einjluss der Umgebung eines Kôrpers au/ die Anziehung oder Abstossung, die er durch eineii Magnet erfàhrt {Poggendorff's Annalen der Physik und Chemie, t. LXXVII, p. 678; 1849). 23o BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. on the magical actions exerted on a body immersed in a magnetic or diamagnetic liquid, arrives at a similar idea. Here is, in fact, the proposition that he states: "The attraction exerted on a magnetic body, which is immersed in a magnetic or diamagnetic fluid, increases or decreases by an amount precisely equal to the diamagnetic repulsion or magnetic attraction that would be exerted on the fluid whose place it has come to occupy. On the contrary, the repulsion exerted on a diamagnetic body that is immersed in the same liquid increases or decreases by an amount equal to the magnetic attraction or diamagnetic repulsion that would be exerted on the fluid whose place it has just occupied. " But Pliicker, in stating this law for the action of a liquid on a magnetic or diamagnetic body which it bathes, refuses to consider the ether as capable of exerting such actions, and thus to explain diamagnetism. He does not want to take into account in his theories forces applied to an imponderable agent, which have no analogue until now. We are going to study the phenomena of a magnetic body immersed in a magnetic medium; the propositions we will arrive at will show us that by admitting the existence of an imponderable medium spread throughout space and susceptible to magnetization, it is possible to explain all the properties of bodies, such as bismuth, to which a negative magnetizing function was first attributed. In the present chapter, we will limit ourselves to establishing the laws of magnetization of a magnetic body immersed in a magnetic medium and subjected to the action of permanent magnets. We will later study the forces to which such a body is subjected. § 2 - Magnetization of a perfectly soft body immersed in a perfectly soft medium. Let us imagine that permanent magnets, designated by index 1, and a perfectly soft body, designated by index 2, areplon CHAP. VII. - AIMANTATION WITHIN AN ENVIRONMENT. 23l The magnetic medium is far enough away to be considered as undefined, which we will designate by the index 3. We shall assume, in the present Book, that this fluid is homogeneous, incompressible and perfectly soft. We shall examine, in another Chapter, whether it is necessary to study fluids that are not perfectly soft. As for the magnetization of non-compressible fluids, we will study it in Book XTL The fluid studied could be a weightable fluid; it could also be the imponderable fluid whose existence is invoked by Mr. Edmond Becquerel to explain the properties of apparently diamagnetic bodies. If we designate by "Ç* the magnetic potential function of the system, and if we keep the notations always used in the previous Chapters, the internal "thermodjmamic" potential of the system we consider will have the value ,f = E(V- TS) -+- W + y e<7 + n i II -Xi ^ Il + ^1 (OlLi)l dv, -^" J L.'-* Il"^1 II J "/ [ï II ''" S Ih ' *"'^''] ''"' V[.î II -'-' S II * '*'<"'- '] *■- The last integral extends to an unlimited domain. We will nevertheless admit that under all circumstances the quantities t? and DlLa cancel out at infinity, so that this integral keeps a limited value. By reasoning on this expression according to the method indicated in Chapter II, we will establish the conditions of magnetic equilibrium on the system. If we assume Fa (OIL2) = , /'^^ , F3 (OIL3) = , J^^ , these equilibrium conditions will be expressed as follows: At all points of the perfectly soft body, we will have ,^=-F2(01Ij) ^, 111.2= - F^CDIU) ^^, e^=.-¥^{d^^) ||. At all points in the middle, we will have Jl,3=_F3(0Il3)^" 'Db3 = -F3(^1L3)?^" a3 = -F3(01L3)^ ux-i oy-i 0Z3 23-2 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. A method analogous to the one we followed in Chapter II will reduce the determination of ..Lo, iti'a, Ca, "1)3, iliïs, 83 to the integration of partial differential equations. This determination will be made by using two certain functions X2 (H'C^), )vs(n\'^), which are nothing other than the expressions of - F2 (OTl/a), - F3 (OIL3), as a function of Ilt.'^ It is not necessary to give here the form of the partial differential equation which the function 'Ç (x, y^ z) verifies in each of the three regions 1, 2, 3. It will suffice to indicate the form of the boundary conditions that it verifies on each of the surfaces that delimit these regions. On the separation surface of the regions 1 e< 3 we have s (x, y, z) being a function known by the data of the problem, according to the equality 5 {x, y, z) = - [ 0.1)1 cos(Ni, x) -+- ill)iCos(Ni, jk) + ©i cos(Ni, z)'}. On the separation surface of the regions 2 and 3 we have (2) [,_ 4^X2 (n^'^)]|^^+[,- 4^X3 (n\';)] 1^ = 0. It would be easy to show, by the method followed in Chapter III, that these equations determine one and only one function "Ç" and that the magnetization defined by this function "v? is a stable magnetization, provided that the magnetizing functions of body 2 and medium 3 are positive and verify the accessory conditions indicated in Chapter III. Let us consider the particular case where the magnetic body and the surrounding medium are both magnetized according to Poisson's theory. In this case, we have F3(DlL3)=-X3(nx:^)=^A-3, A "2, A "3 being two constant magnetization coefficients. Let O,, ÏD^, 1^3 be the magnetic potential functions of the three magnetized regions I, 2 and 3. We will have \.= t5i4-t)2+ W3. CHAP. VII. - AIMANTATION WITHIN THE ENVIRONMENT. 9.33 Since the function i:)< is known from the problem data, we only need to determine the function This function is harmonic in each of the three regions 1, 2 and 3. On the separation surface of regions 1 and 3 it verifies the equality (3) _^(, + 4"/")j^^ = o. On the surface of separation of the regions 2 e^ it verifies the equality (4) (t+- 4-1^^-2) j^ -+-('-+- 4 T^^a) ^ =o. According to Maxwell (') and É. Mathieu (2), this function ^^i^) is identical to the magnetic potential function of a mass occupying the place of body 3, immersed in a non-magnetic medium, and whose magnetization coefficient k would be given by the equality H-47:/.2 I -h 471/1 = - - - 1 -h- 471/13 But this would only be correct, as is easy to see, if equation (3) were replaced by the equation The proposal of Maxwell and É. Mathieu cannot therefore be maintained. The equations of the magnetic equilibrium on the system can be written ^2 = - Fa (OR 2) Dl,2= - FaCDRî) dy ^ ^n . a. = -F2(0rc2) O _ T7 /, 3 be the components of the magnetization at the point (x, y, z) of this fluid; let dv be the volume of an element of this fluid traced around the point (^, jKî ^)- ^this element is subjected to forces foreign to magnetism having as components pXdv, p'V dv, pZdv, being the density of the fluid at a point of the element dv, and X, Y, Z being certain functions of (^, j', z). Let us suppose first that the element dv has the shape of an infinitely small sphere, having for center the point (x,y,z)', let us imagine that this small sphere, dragging with it its magnetization, experiences an infinitely small rotation around its center. In such a displacement, the forces extraneous to magnetism do no work. If therefore the fluid is in stable equilibrium, we will have, for any virtual displacement of this kind (i) ^f-o and (2) o^f'>o. The rotation of our small sphere makes the components of the magnetization at the point (x,y,z) vary by 8 l-, 5itî>, oS. We have therefore, by designating by %*) the magnetic potential function of the whole system, O-f = -- O'X H Olll) -f- -T- oc- H r-- OJII OTC. \^ox Oy ôz odK J But the quantities o-l., oill", o3 are not random. The small sphere drives its magnetization whose size does not vary, which gives oOPl = o or (3) 5.1> oX -f- Uî) o\li) + S o3 = o So we have (4) \ Ox oy OZ d,, and, by means of relation (3), this quantity must become equal to o. 238 BOOK IX. - KT TIIERMODYNAMIC AIMANTATION: . According to a known principle of the calculation of variations, it is necessary and sufficient, for it to be so, that we can find a function 0(^, y, z) such that we have, at any point of the guide, [ol,-hO(^,j,5)^']8.l.>, whatever Ovl., ôilï), oC, which requires that there exists a fono tion 8(:r,jK, z) such that we have, in any point of the fluid, (5) |ill,=.-6(a-,j',s)^\ A magnetized fluid cannot be at rest unless the magnetization is tangent to the line of force at each point. From equality (4), we deduce -^ C>2 1)1) + -^ B^ê = f-^^ij{,-i- I21^2^i,_|_ ir52£) ^t, OR, by virtue of the equalities (5), 82 i = - - - ' ( .1, S2c.i 4- 0)!, 52 1)1, -^ 3 52 S ) dv. Moreover, equality (3) gives Jl,o2 "l, + Dbo2ill, -+- 8o2£ -t-(oX)2 4-(oDl,)2+(o3)2= o, which allows to write and transforms inequality (2) into (6) ^{x,y,z)>o. The function 0(^, y, 5) is necessarily positive. CHAP. VUI. - INCOMPREHENSIVE FLUID WITH AIMANT. îSq This result immediately highlights an important proposition. The equalities (5) give each of these two integrations extending to the entire volume of the fluid. Let's assume that the system is devoid of any permanent magnet and contains no other magnetic body than the fluid. In In this case, the quantity / U^;r- d*^ represents the magnetic potential of the system and is essentially positive, unless the magnetization is zero at any point. The same is true, by virtue of /OU T- T\-lV' Equality (7) is therefore impossible, unless the magnetization is equal to o at any point; hence the following theorem: Any magnetic fluid, subtracted from the action of magnets, cannot be at rest if V magnetization is not, at any point, equal to o. These first theorems obtained, we will ask ourselves the following question: Inside a magnetized fluid, we draw a closed surface S. The various elements of the fluid contained in this surface are subject to certain given forces, foreign to magnetism; these forces, which would remain if the magnetization at each point of its elements were reduced to o, are of internal origin to the fluid mass enclosed by the surface S or of external origin. Let (8) ç>\dv, pXdv, pZdv the components of the external force of this kind applied to the element dv of density p. The magnetized bodies outside the fluid contained in the surface S also exert forces on the various elements inside the surface. If xJP(.r,j, 5) is the magnetic potential function of those magnetized bodies outside the surface S at a point {x^y^z) of the element dv inside the surface S; if ,.1., Kl>, a are the com :>.40 LIVHE I\. - MAGNETIZATION AND THERMODYNAMICS. (^,^, z), in any modification where each element of the dç will move, dragging with it its magnetization, the forces in question will perform an elementary work which has the value (y) cB -iA= dx \% dz ) dv, the integration extending to the whole volume enclosed by the surface S. Let us imagine that, without modifying anything to the external forces which are listed here, we remove the obstacles that the presence of bodies outside the surface S opposes to the deformations of the fluid that this surface contains. Will it be possible to maintain in equilibrium the fluid, supposedly incompressible, which the surface S contains, by means of suitably chosen forces, applied to the various elements of the surface S? To solve this question, which constitutes the fundamental problem of hydrostatics for magnetized fluids, we will use a method similar to that used by Mr. J. Moutier(*) to study the classical problem of hydrostatics. On the surface S {fiig- 20), consider two elements which FÏK. 20. conks AB or ofS, and A'B' or dS' . The former is subjected to a force OP or P dx Us (23) / [ ( t/AFCIlA' L dx âx dx \ dx dy /d^ d^ \ dx dz (/s d^ ()lft) dy dx d^ d^o dy dy dy dz à^ d^ dz dx d^ d3 dz dy d\9 dj. dz dx dy ") -] If we now ask (-24) -<) = XJJ) -f- 10, t? being the magnetic potential function of any acting Taimantalion, the equalities (i i) to (24) will allow us to transform equality (10) into / dx (15) [cos(P, N)^lang(P, ^)sin(N, D)cosn]p-+- e.U r -^1 II d<^ - [cos(P', N')^- tang(P', N')sin(N', D')cosN'J P' - 1| -X, - + f__] [?(X+^) 'afgha L p(z+:)+"^"-^" &Ç dX dx dx d^ dX dx dy 0^ toX dx OZ dx ||,/s' dp toei ^^ dz dx J "^ f 1 dy dz dy j -^ &Q d^ dy dx d^ dJU dy dy dp d\\\, dp à^l , -- -■ \ dz dy dz dz dz J This equality must take place whatever the shape of the infinitely loose channel which connects, through the fluid^ the two elements dS and dS'. It must therefore remain true if we replace, in the generatrix AFGHA' of this channel, the arc FGH by the arc FKH. It is not not difficult to conclude that the quantity under the sign / must be the total diférential of a function of :r,y, xî, continuous and uniform at any point of the fluid. Let us assume in particular that the fluid is homogeneous and incompressible, so that p has the same value at every point of the fluid. Let us suppose, moreover, that the forces extraneous to magnetism admit a potential function, i.e. that there is a function Q, -i46 BOOK IX.. - THERMODYNAMIC KT MAGNETIZATION. continuous and uniform inside the fluid such that we have of the \x \ ' "^ dz Finally, let us take into account the equalities (5), which must be linked for the equilibrium to be possible. The quantity under the sign / can be written I r / ^.H, rir^a. "^ -(- G - - I dx \i{x,y,z)\_ ôx dx I =il. - -1- llî) 3-^) dy . ci/ âj ày ) -' or, by noticing that X dx ^ lll) d\\\y + 8 c?S = .m d^\l . (27 ) -pdii- ,.^ '^^^ . "?orL . Ù{x,y,z) For this quantity to be a total differential, it is necessary and sufficient that the function Q(.r, i', z) does not depend on x, r, z, except through the magnetization intensity OR. Let us therefore posit (28) %(x,y,z)==^e(D]L). The equalities (5) will become dx (29) ^,)î,=_e(oiL)^, If we remember that, according to the inequality (6), the function ô(.r, j-, z) and, therefore, the function 6(01i) is positive, we arrive at the following result: When an incompressible, homogeneous, magnetized, non ICVP. VIII. - INCOMPRESSIBLE MAGNETIZED FLUID. 247 or not of coercive force, is at rest, the magnetic particles present exactly the distribution they would present in a homogeneous body which would have the same shape as this fluid, would be subjected to the action of the same magnets, would be devoid of coercive force, and would present a suitably chosen magnetizing function 0(O1L) If the fluid is devoid of coercilive force, the comparison of the equalities (29) with the conditions of magnetic equilibrium shows that the function 0(O1L) is determined and equal to the magnetizing function F(0\u) proper to the fluid. If, on the other hand, the fluid is not devoid of coercive force, nothing in what we know so far determines the function 0(Orc). We shall see, in § 3, the importance of this remark. § 2 - Pressure exerted by a magnetized fluid. Let's go back to the equality (aS). The quantity under the / sign must be the total dillerential of a function of x, )', :; finite, continuous and uniform inside the fluid. Let us denote by - n(.r,j'j ^) this function. Equality (25) will become (3o) ( [cos(P, N) - tang(P, N) sin(-\, D; cosi\ J P -^ I _[cos(P', N')^tang(P'. N')sin(\', D')cos^'Jp' - C/ v> -, - OX 1A9 - ax - n' = o. The directions D, D' are arbitrary. The previous equality must therefore take place whatever the quantities sin(i, D)cosl\, sin(N', D') cos i> . This requires that we have tang(P, N) - o, tang(P', N') - o. The forces that must be applied to the various elements of the surface S to maintain the fluid in equilibrium are therefore normal to the surface S. 2 j8 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. Let us denote by P t/S the normal force applied to the c/S element, completing it positively when it is directed towards the interior of the fluid. Equality (3o) will become dx _^n - P'- ùx n'= G. Let us denote by OIl and Oit' the magnetization intensities at a point of the elements 2-^-. Let Z be the distance to the HfF plane of the liquid level in branch B; let Z' be the distance to the HH' plane of the liquid level in branch B'. At a point on the level surface in branch B, we will have P2^Z + Ai(;)Hi) - "F2(;)H2) --- Cl,. At a point of the level surface in the branch B', we have Hence, by subtracting member from member, ( 40 ) p, ff{Z - Z')=^ W, ( on 2 ) - U% ( iVil', ) - A, ( ;)H , ) ^ A, ( DM \ ). It will thus be established in general, between the two branches, a difference of level given by the preceding equality. In particular, let us assume that the branch B' is located in a region where the magnetic field is not very intense, so that in this region the fluids are not very magnetized. The previous equality will be reduced to (40 p,g{Z~Z') = W, ( ;)rt2 ) - A, ( OK 1 ). CHAP. VIII. - INCOMPRESSIBLE MAGNETIZED FLUID. 255 If the quantity V2(;)lL2)-A,(Dlli) csl positive, the liquid will rise more in the magnetized branch than in the other branch; the opposite will happen if the quantity ^i'-2(0li2)-A,(;^ll,) is negative. Let us assume, for a moment, that the aether is also devoid of coercive force, and that the two fluids magnetize weakly and according to Poisson's theory. Equality (4i) will then become Piff{Z-Z )=r-- ~ l\àxj '^[ôfj '^[dzj J Pliicker (' ) has given experiments which agree with this relation; Quincke has used it to determine the excess of the magnetization coefficient of a fluid over the magnetization coefficient of the elher. This method loses its value if, as the experiments of M. Joubin seem to show, the elher is not devoid of coercive force. (') Plucker, Expérimental Untersuchungen iiber die Wirkung der Magnete auf gasfôrniige und tropfbare FLiissigkeiten ( Poggendorff's Aniialen, l. LWIII, p. 549; i848). 256 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. CHAPTER IX. ACTIONS EXERTED ON NON-MAGNETIC BODIES. § 1 - Basic formula. Let us imagine a body of low magnetism immersed in a medium of low magnetism; let us suppose these two bodies devoid of coercive force; let us subject them to the action of powerful permanent magnets. Such a system lends itself to a very complete and elegant study, the principles of which have been indicated by Béer (' ). We will explain here this study, the conclusions of which allow the interpretation of a great number of physical phenomena. Some of the propositions we are about to arrive at can be extended to the case where the environment is supposed to be provided with coercive force; but we will not stop to discuss this case. Let us assume a perfectly soft body 2 immersed in an unlimited medium, devoid of coercive force 3 and subjected to the action of permanent magnets 1. If O), Oo, ^s are the magnetic potential functions of the bodies 1 , 2, 3, the internal thermodynamic potential of the system, supposedly deprived of electricity, will be 1 ij W ôxx -i.J II ÙX-i ij ,. 1 , m J ^ g' ) 1+1'):.) Il ; ^rlk <^(Wl+t%)||, (i) + / U^ ^- UA-2+/ Ul.3 . \dv^ 1 ..-'11 "■* 2 I I J W "■^'i I I f -+■ I ^i{ 0)Vi ) dvi + I S\ ( Oli 2 ) dvi -f- j $i{ on 3 ) dvi (' ) Béer, Einleitun g in die Elektrostatik, die Lehre vont Magnetismus und die Elektrodynamik. Lehre vont Magnetismus, Cli. VI, p. 2i3; Brunswick, i865. CHAP. IX. - BODIES OF LITTLE MAGNETIC FORCE. 267 Let us assume that body 2 and medium 3 are very weakly magnetic; we can, as we saw at the end of Chapter VII, neglect, in the expression of §, the terms / dVi, f dvi, / dvz, f atxs di>3. We can also write (2) A>,^-F,(DXL,)^^, oA "3 - -FsiDTLz) i)î" = _F2(D1L0^' llîja^ -Fs(DXis) Q,=-F,{D\L,)^-^, ©3 = -F3(D1V3) atxs equations which express, as we have seen in Chapter VII, that body 2 and medium 3 magnetize as if each of them existed alone in the presence of permanent magnets. If we then assume, according to equality (lo) in Chapter V, W.OIL3) = equality (i) will become -? = E(r-TS)-^y F.,(Jrc2) FaCOlU) ^T^3(01i3), .1,, dXx dvi ■Js.ii DIV 1 ) dvi - r ¥2 ( DU 2 ) ^i^2 - r ^3 ( Dit 3 ) dv3 Let us suppose that body 2 is moved infinitesimally little in medium 3, and that at the same time the magnetization of the body and of the medium is varied in such a way that the conditions of equilibrium remain satisfied; let us suppose, moreover, that this modification changes nothing in the physical and chemical state of the various elements of the system. Under these conditions, the given forces, internal to the system and foreign to magnetism, perform a work d^i given by rf5,=- ES(r - TS). D. - II. 17 258 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. The quantity does not experience any variation. We have therefore (3) 8#=:_c?S/- 8 fWi(D]U)di>^-^ f ^2{DTi3)di^3. This very simple formula is susceptible to a slight transformation. The formulas (2) define the quantities X^^ 1IÎJ3, 83, ^3, even for a point (.To, y^^ 2^2) of the volume occupied by body 2. The quantities thus defined are the components and magnitude of the magnetization that would occur at the point (.2^2? J'2j -^2) if, with body 2 removed, it were replaced by a fluid of the same nature as that filling space 3. Consider the quantity It represents the value that the quantity would take J^'3(0]U)dv, extended to all the space outside the permanent magnets if we assume that this space is filled exclusively with the fluid 3. This quantity has a value independent of the position of the volume 2; it does not vary when we move this volume. We have therefore 8 r ^3 (0103)^^3 -1-8 r ^3 (31^3)^^2=0, which allows to replace equality (3) by the following one: This is the fundamental formula on which the theories we will develop are based. CHAP. IX. - SMALL MAGNETIC BODY. iSg § 2 - Edmond Becquerel's theorem. Equality (4) shows that the work of the magnetic actions that tend to move body 2 can be seen as expressed by the equality (5) dr = ci f [W,{D]l'i)-Ws(DJL3)]dv.2. If body 2 were placed in the position it occupies in relation to the permanent magnets, without being surrounded by the medium 3, the magnetic actions tending to move this body would correspond to a virtual work whose expression, found in the same way, would be If the volume 2 were filled with the fluid 3 and the space surrounding it were empty, this mass of fluid 3 would have, at each point, the magnetization designated by Df^s in formula (5). The magnetic actions tending to move it would efi "ect a virtual work d-cs'-^o Cw3iDK3)dvi. Equality (5) can therefore be written dx =^ dxi - . - ) C^3 - - A'3 -- . 0Z2. Ozz this equality becomes / , / ,()(X')i-l-'0,+ t)3) , , , ^aCtOi + lO-H-lOs) ^' + 4-^2) ^^ +(. + 4-A-3) ^^ =0, or, by noticing that ^3 and "O^ are very small and that f^Na ' (JN3 = 0, This condition requires that - - > -r - ? -r - have, as a hindrance, some values inside the small magnet body; what - j -r - ? -r- ^ also have finite values in the immediate vicinity of this small body, these values becoming infinitely small at a finite distance from the small body. Faraday's law does not therefore extend, in general, to even very small strongly magnetic bodies. There is, however, one case where, as Sir W. Thomson (' ), this law extends to a very small body, even if it is assumed to be strongly magnetic; this is the case where the body has the shape of a very small sphere. (') W. Thomson, On the forces experienced by small spheres under magnetic influence; and on some on the phenomena presented by diamagnetic substances {Cambridge and Dublin mathematical Journal, May 1847). "~ Po-pers on electrostatics, art. XXXIII). CHAP. IX. - NON-MAGNETIC BODIES. 269 Let us neglect, in fact, the variations that the field created by the permanent magnets presents in the very small extension of this sphere; let us also neglect the influence of the very non-magnetic medium that surrounds it; the components of the magnetization at a point of the sphere will be [Book VIII, Chap. II, equalities (7)] f\D2 ^^ lft)9 k. dXDi I-+- ^1T>^2 dxi k. aoi 1+ 5Tt>t2 tof. k. dtDi i-t-lTrA-2 dzi According to the general laws that give the forces acting on a magnetized body [Book VII, Chap. I, equalities (12)], we see that the magnetic actions exerted on this small sphere will be reduced to a single force having as components "" -A I J W-^2 àxl CJ72 àx^dji dz2 dx^dz^J '^' 1+ -^T^f^i I -T- -TZKi ~ 4 7 J \ (^^2 to^2 to^ï "^^^2 <^J2 toz% toZ2 dz'l ) ^ ^' or, by designating by R the radius of the small sphere. Y = 2 tcA^o ^ (^ \ I K" 1^ llUi, I-l ^./:2 -2 ^A "2 ^ ô -7 K* ^ bed),, j + \-Kk, 2 3 Tr/C2 ^ K' It V)i. 4 . <>5 71 A" 270 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. These equalities, of the same form as equalities (8), justify the extension of Faradaj's law to a small strongly magnetic sphere, in the limit where the magnetization coefficient of this small sphere can be considered as constant. Consider a permanent magnet and a small sphere of soft iron placed in contact with this magnet. The magnet will exert on this small sphere an action whose component along the normal N^ outside the magnet will have the value ?>''^' d N= ^ R3 nOi. a To pull out this small sphere, it will be necessary to apply to it, in the direction of the normal N^, a force F= - 2 R3 j_r/£ËiV^/^>V d^e [\dx J ' \dy ) 4 , ^Ne \dx/ \ ày I ' \ dz By determining the value of this pull-out force at the various points of the magnet, we will have numbers proportional to the values of Jamin ('), who made many determinations of this kind, believed he was obtaining numbers proportional to and, consequently, likely to be used to determine the fictitious distribution equivalent to the magnet. It can be seen that the pull-out method is unsuitable for this purpose. (') Jamin, Sur la distribution magnétique {Comptes rendus, t. LXXV, p. 1672; 1872). - Sur la distribution dif magnétisme {/bid., p. 1672). CHAP. X. - MAGNETIC SPECTRA. 27 1 CHAPTER X. MAGNETIC SPECTRA. § 1 - Spectra formed by weakly magnetic powders. When a sheet of paper is placed in a magnetic field and sprinkled with iron filings, the filings form trails that constitute a magnetic spectrum. The shape of these trails is given by a very simple rule: draw the level lines, i.e. the intersections of the sheet of paper and the level surfaces of the field; the trails of filings spread on the sheet of paper will intersect orthogonally these level lines. M. E. Colardeau (' ) has shown that if a very intense field was used and if iron filings were replaced by powders that were not very magnetic, such as crystallized, silky and friable oligo-iron, red iron oxide, oxide of the battitures, various oxides of nickel and cobalt, the streaks that constitute the magnetic spectrum would have a different shape. M. Colardeau thinks that the powder trails draw, in this case, the level lines themselves; we shall see that this is not the case. We will start by studying these spectra formed by poorly magnetic substances. Let t3 be the magnetic potential function of the per (") E. Colardeau, Sur les spectres magnétiques produits au moyen de substances peu magnétiques {Journal de Physique, 1' série, t. VI, p. 83). 272 HVBE IX. - MAGNETIZATION AND THERMODYNAMICS. manent, ©2 the magnetic potential function of the powder grains, 'Og the magnetic potential function of the medium. If the powder and the medium are not very magnetic, the potential functions t^a and TO3 will be very small compared to t!),. Under these conditions, a parcel of volume v.^ will be subjected to a force whose components will be given by the equalities (8) of the previous chapter: x,= h -k, 1 dxi '^1'. Yo - kl -k. atyt Vl *2 - 2 z" = k^ -h. (^nio, fo, Let us assume that [k'i - A3 ) is positive and that the parcel is subject to sliding on the sheet of paper. It is easy, by the previous formulas, to find the direction of the force that tends to put it in motion. Let us draw, on the sheet of paper, the isodynamic lines, that is to say the intersections of the sheet of paper with the isodynamic surfaces of the magnetic field created by the permanent magnets. The magnetic force tangent to the paper that tends to move the particle V2 orthogonally intersects the isodynamic line on which this particle is located. If, as we shall assume, (A "2 - k^) is positive, in which case the particle is paramagnetic, this force is directed to the side of the isodynamic line where the field has a greater intensity than on this line itself. Let us now imagine that the sheet of paper presents an obstacle: either an asperity of the sheet of paper, or magnetic particles that some cause has stopped. It is clear that any magnetic particle that comes to bump against this obstacle, being on the side of the obstacle where the field intensity is lower, will be stopped in its movement. It will itself form a new obstacle against which new magnetic particles will come to rest; each obstacle will thus be the starting point of a tramline, located on the side of this obstacle where the field intensity has the least value and whose general direction will be that of the normal to the isodynamic line passing through the obstacle. CHAP. X. - MAGNETIC SPECTRA. 278 Thus, in a spectrum formed with a weakly magnetic substance, the streaks that the powder forms on the paper orthogonally intersect the isodynamic lines. § 2 - Spectra formed by strongly magnetic powders. It is much more difficult to study rationally the spectra formed with a strongly magnetic substance. We will only be able to do this study in the case where the field created by the electromagnets is strong enough to magnetize the particles of filings to saturation. We shall further assume that, although the field strength is very large, the variations of this strength remain finite as one passes from one point of the field to another, and that they are infinitesimally small in vm domain whose dimensions are comparable to those of the filings. , . , dXD^, dïO, dXlli . , ,. , , The quantities -r-^" t-^j --^ will be, at any point, negligible . d-Oi dto, d Oy . 1 f - -Il with respect to -^-^ -^ - ' ~Jr * V^^'^'- ^ 1^ the magnetic potential of the medium, there is no need to take it into account. The JU, ii!>, 3 components of the magnetization at a point of a soft iron particle are given by [Chap. IV, equalities (i3) and(i4)], / a _ P ^^"^i (0;'it^2-- j-^, P being a constant. Let \'^2 be the magnetic potential function of the other particles of filings, and 1JP2 Aa magnetic potential function of the magnetization distributed on this particle, whose volume is v^. When this particle is moved, the magnetic actions applied to it do work equal to the sign of the variaD. - II. 18 274 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. tion of the quantity of the /("-^'^^ \)!)2 ^ i- S2 -.; - ) dVi <>y% toz^ Equalities (i) give us DU 2 := P. The quantity OlLo having an invariable value, we have / rf 2 (011 2 ) C?(''2 - O. Let 3.37, B^K, Ss be the components of the translation of the particle V'i'i Sx, Su-, Sv the components of the rotation. The variations experienced by the first and third terms of expression (2) will in general be of the order of v^'^x, P2SJK, ..., v^ht. The same will be true, in general, of the variation undergone by the second term. However, there is, to this last statement, a case of exception. Let us suppose that the particle ç^o is located at a very small distance from another particle p^ ^^ of the same order of magnitude. According to what we saw in § 4 of the previous chapter, the quantities -^, -^, -r- > which have finite values in immediate contact with àXï 0J2 dZi * the particle p^? take very small values of the order of v'.^ at a finite distance from this particle. So, in the vicinity of the .1 , , ,- , ■f.d'Çi ^dVi ^d-^i i j 1 particle v^. the quantities 0- - , 6^^ - ? 0^-- have very r 25 T values ^x the magnetic potential function of the other magnets. The only variable part of the internal thermodynamic potential § is the quantity I\ dx the integration extending to the moving magnet. In this magnet, magnetism will be assumed to be independent even of temperature. Therefore, the above quantity does not depend on the absolute temperature T. Equality (2) is reduced to E^Q-^-O^-^-^^Ce-ô/l dx dv. Besides, we have, in the conditions where we are placed, 'J\ dx di> = - dGi, d<£i being the work done by the actions of the stationary permanent magnets on the moving permanent magnet. Thus we have E f/Q + 8 y - -= rfSe -+ d^i, equality that could have been foreseen by the principle of displacement without change of state. 2" A perfectly soft body, not immersed in a magnetic medium, is placed in the presence of permanent magnets. Bonds hold it immobile. At a given moment, the bonds are removed. The body rushes towards the permanent magnets. In its race, it meets a calorimeter where its living force is damped. How much heat is given up to this calorimeter? aSo BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. Let A be the initial state of the system and B the final state. In the initial state as in the final state, we have ■^ mv Equality (2) therefore gives d^\ di The physical and chemical state of the soft iron being the same in the initial state and in the final state, we have f^K- ^B = ^A-.%+ IJriiDTL, ^)di']- If^iDXl, T^)d^]. So we have EQ = ijA - -Tb -^/[^( ^^l^A, T ) - T -/[''' DKb, T)-T (j.T(,mA, T) ^.f(;)iLB. T) atT dv V'^l! tithing. In particular, let us assume that the initial position is very far from the permanent magnets, so that we have approximately OÏL A = u. We will also have significantly ().T(OKa,T) dv = o, yri(3iu, T)-T dv. So we have (4) EQ=- riiju^ dv-- rik - f\§{d\i, T) - T ôx ()g^(,m ,T) \¶dv V'-^I' d(ô.. The first three terms refer to the final position of the magnet. CHAP. XI. - HEAT AND MAGNETIZATION. 281 We have seen [Chap. V, equality (i i)] that we have rii ,.^|L/,_.i r||.i/-,^'|L/.-^ rs(DXL,T)d. j II t^-^ Il ^'j II à^ Il j = - g'- Cnvciv- fwi ;)ii , T)dv, the first integral extending to the whole system and the second to the soft iron; the function ^(Oli, T), defined by the equality (lo) of the same Chapter, is positive for magnetic bodies. On the other hand, /'Says ;^n So we have lEq=-^fu\Ddv-hfw{D\L,T)dv r - T/ / .,, .,., ,p -TiT^ - 'dDYidv-^ I dise. This equality shows us that, if the magnetizing function decreases, remains constant, or increases slightly as the temperature increases, the heat released exceeds the heat equivalent to the work of external forces. If Poisson's theory is applicable to the considered perfectly soft body, we have '^^^^^' ^^ ^ -ilrri' ''^^"" '^^ " ^'^^' and the previous equality becomes (6) EQ = ^/n O * + 1 [^L_ .__ T il j(V)]/^'^' * -X"'"'" 3" A perfectly soft body 2 is immersed in a magnetic medium^". It is subjected to the action of V permanent magnets 1 and of any external forces. H moves without experiencing any friction. We suppose that the magnetic distribution is, at each instant, on the body and in the medium, that which would be appropriate for equilibrium if the body were stopped in the position it occupies at that instant. What quantity of heat is, at each instant, released by the system whose temperature remains invariable? 282 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. If we denote by d(Bi the work done by the internal forces during the time dt^ we have 8 \ - - = "?Ge-+- d^i, and equation (2) will become (7) E"?Q = -8(j-T^|)-.ZG,. Let us denote by ô, § the variation that the internal thermodynamic potential would undergo during the time dt^ if the parameters which fix the position of the various parts of the system varied alone, the other parameters, which fix their magnetization, their physical and chemical state, remaining invariable. Let us designate by 60 3" the variation that the internal thermodynamic potential of the system would undergo if, on the contrary, these last parameters varied alone. We will have, on the one hand, and, on the other hand, the principle of unchanged travel state gives us d(li= - 81 J. Equality (7) will thus become E^Q--82eF-4-T8^. But, at each instant, the state of magnetization of the system is precisely that which is appropriate for equilibrium. We have therefore, at each instant, 62^ = and, therefore, (8) E^Q = T8^, equation of the same form as equation (3) which is suitable for a reversible phenomenon, although the transformation considered at this moment is feasible and, therefore, not reversible. Since the magnetic potential does not depend on temperature, we simply have 4 =/-^4l^''-'--/^^lg^'^-=- Besides CHAP. XI. - HEAT AND MAGNETIZATION. 283 dD]l, F3(01L3,T) Equality (8) thus becomes Let us first assume that the medium is non-magnetic. Equality (9) will simply be If the variations of the magnetizing function are in the same direction as the variations of the temperature, the body will release heat when its magnetization decreases and will absorb heat when its magnetization increases. VinKcrse will take place if the variations of the magnetizing function are not in the same direction as the variations of the temperature. Let us then assume that the body and the medium are both weakly magnetic and that the magnetization of both is given by Poisson's theory. We will have first, denoting by A "2(T), ^^(T) their magnetization coefficients, F2(3rt2, T) = A-2(T), FsC^Ks, T)=/f3(T). We will have, in addition, 01i| = X:|(T)nt),, 01Ii=Â:l(T)ni!),. So instead of equality (9), we have equality Now, in the whole space {y 2+ ('3)) the function lO^ is harmonic. If therefore we denote by t/Si an element of the surface of body 1, by N3 the normal to this element towards the interior ^84 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. of the middle 3, we will have Tn t), dv^ + ÇwOx di>3 + § X')i ^ ^S, = G. It is easy to deduce from this that, in any displacement of the mass 2, we have where fu tOi dv2 -t- Tn t)i dvs = o. Equality (lo) thus becomes (n) E"?Q=-^-^[^2(T)-A-,(T)j3ynt5,^P,. Let V be the volume of body 2. Let J^ be the mean value of the square of the field intensity in the space occupied by body 2. We will have J2V= fnVidi>i. The equality (i i) thus takes, in the final analysis, the following form This equality can be stated as follows: A magnetic field, generated by permanent magnets, is filled by a non-magnetic medium in which a non-magnetic body is immersed. This body is moved in such a way that the mean square of the field intensity at the various points of the space it occupies increases. If the excess of the magnetization coefficient of the body over the magnetization coefficient of the medium decreases when the temperature increases, the modification causes a release of heat. It leads to an absorption of heat if the same excess increases with the temperature. The results we have just arrived at were stated in 1878 by Sir W. Thomson (<). In i883, Messrs. E. Warburg and (') Sir W. Thomson, On the thermoelastic, thermomagnetic and pyroelectric propei-ties of malter,%l {Philosophical Magazine, 5" series, t. V, p. 4" 1878). CIIAP. XI. - HEAT AND AIMANTATION. u85 L. Honig (*) represented them by a formula which is not exact. The first exact demonstration was given in 1889 by M. Paul Janet (2). § 2 - Specific heat of a magnetized body. Several authors (^) have studied the influence that magnetization has on the specific heat of iron. It is easy to study this influence for the specific heat under constant volume. This one, in fact, is linked to the internal energy by this very simple relation nj being the mass of the body. If we refer to equality (i), this relationship becomes According to a transformation analogous to the one we have made the equality (4), if we designate by y the specific heat of the non-magnetized iron, we have ( Ev^ir -^i) ^- ~^_ ^ j'nV dv - ~ fw(DK , T) di> (') E. Warburg and L. Honig, Ueber die Wàrme, welche durch periodiscli wechselnde magnetisirende Kràfte im Eisen erzeugt wird ( Wiedemann's Annalen, t. XX, p. 8i4; i883). (') Paul Janet, 5m/' la chaleur de combinaison du fer dans un champ magnétique et sur les phénomènes thermomagnétiques {Journal de Physique, ï" série, t. VIII, p. 3i2; 1889). (') J. Stefan, Ueber die Gesetze der elektrodynamischen Induction {Sitzungsberichte der Akad. d. Wiss. zu IFi'en, LX[V, 1' Abtlieil, p. igS; 1871). - A. Wassmuth, Ueber die specijische Wàrme des stark magnetisirten Eisens und dos mechanische ^Equivalent einer Verminderung des Magnetismus durch die Wàrme { S itzungsberichte der Akad. d. Wiss. zu If ic/?, LXXXV, 2" Abtheil., p. 997; 1882). - Ueber eine Anwendung der mechanischen Wàrme- théorie auf den Vorgang der Magnetisirung {Ibid., LXXXVI, 2" Abtheil., p. 539, 1882). - Ueber die bei Magnetisirung erzeugte Wàrme {Ibid., LXXXIX, i' Abtheil, p. io4; iSS'i ). - J. Stefan, Ueber thermomagnetische Motoren {Wiedemann's Annalen, Bd. XXXVIII, p. 427; i^ a86 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. Two cases are to be distinguished: 1° Do we want to calculate the true specific heat of magnetized iron, i.e. the quantity c such that c dT represents the heat that must be supplied to the iron to raise its temperature by (TV) without modifying its volume or its magnetization? The derivatives with respect to T which appear outside the signs / are then true partial derivatives 5 we have ^ / Ut") dv=-.o ..fnv and, therefore, If the body follows the Poisson theory, we have W(0]\. T) = . ^ ' ^ ->./t(T) F(Dll, T) = /c(T), and the previous equality becomes If the body is not very magnetic, we have orL2 = A-2(T)n\v, and equality (i3) becomes (i36a) E.(o-Y)=-j-^-T^A_J^;^[-^J ynxui.. 2" On the contrary, do we want to suppose that c dT is the quantity of heat necessary to raise the temperature of the iron by dT, its magnetization varying because of the modification that the change of temperature brings to the magnetization coefficient. In this case, the differentiations with respect to the temperature that appear outside the / signs in the equality (12) must be taken taking into account the influence of T on the magnetization . The result becomes much more complicated. CIIAP. XI. - HEAT AND MAGNETIZATION. 287 Let's assume that the body is weakly magnetic, so that we can neglect the term (jui is of the order da / 0]1'^ dç, while the others, if we admit the Poisson theory, are of the order of 7-777- - Let us note in k{i) ^ in addition to the fact that W(OR,T)=^-|g-) = ^n... F(OR,T) = ^(T), and equality (12) will become The two values of c, given by the equalities (i3 bis) and (i4)) are different from each other; they differ from those proposed by the authors. As for the study of the specific heat under constant pressure of iron placed in a magnetic field, it is so complicated that we do not want to discuss it here. This would be the place to talk about the influence that magnetism exerts on chemical phenomena; this influence, highlighted by the experiments of Mr. Ira Remsen (') and Mr. Rowland (^), has given rise to the theoretical research of Mr. Th. Gross (^), (' ) Ira Remskn, Chemical action in an electric field {The Electric Light, vol. IV, p. 126; 1881). - H -V. Jueptner, L'in/luence du magnétisme sur les métaux au point de vue électroly tique {La Lumière électrique, t. X, p. 469; i883). (') FoiV Oliver J. Lodge, Sketch of the principal electrical papers read before Section A during the late meeting of the British Association at Manchester 1887 {Electrical Review, 28 September 1887). (') Th. Gross, Ueber eine neue Entstehungsweise galvanischer Strôme durch Magnetismus {Sitzungsber. der Akad. d. Wiss. zu Wien, XCII, 2''Abtheil. p. 1378; i885). - Ueber die Verbindungswdrme des magnetisirten Eisens {Verhandlungen der physikalischen Gesellschaft zu Berlin, 22 April 1887). 288 BOOK IX. - MAGNETIZATION AND THERMODYNAMICS. of Mr. Nichols(*), of Mr. Paul Janet (2). But we would be afraid, by exposing here the theory of these actions, to extend too far the dimensions of the present volume. We will therefore ask the reader to refer to the works we have just quoted and to what we have said elsewhere (^) on this subject. We will not speak of the changes in concentration that the proximity of a magnet causes in a dissolution containing a magnetic salt. We have devoted to the study of this question a Memoir (^) to which we will refer the reader. (') NiCHOLS, On the chemical beaviour of iron in the magnetic field {Sillimann's Journal, 3' series, t. XXXI; 1886). (') P. Janet, De l'in/luence du magnétisme sur les phénomènes chimiques {Journal de Physique, 2° série, t. VI, p. 286; 1887). - 5m/' la chaleur de combinaison du fer dans un champ magnétique et sur les phénomènes thermomagnétiques {Journal de Physique, 2" série, t. VIII, p. 812; 1889). {') On magnetization by influence, Chapter VI, § 2 and 3 {Annales de la Faculté des Sciences de Toulouse, t. II, p. D.98). (*) On the dissolutions of a magnetic salt {Annales de l'École Normale supérieure, 3' série, t. VIII, p. 289; 1890). BOOK X. THE MAGNETIZATION OF CRYSTALLIZED BODIES. CHAPTER ONE. EQUATIONS OF THE MAGNETIC EQUILIBRIUM ON CRYSTALLIZED BODIES. § 1 - History. We have studied in detail the problem of magnetization by influence of isotropic bodies; we now have to return to the general case of this same problem, to the study of magnetization by influence of any non-isotropic bodies, and in particular of crystallized bodies. Poisson had indicated the characteristics that distinguish the magnetic induction of crystallized bodies from the magnetic induction of isotropic bodies twenty-three years before any experimenter thought of verifying his predictions. We have seen (Book VIII, Chap. I, § 1) how Poisson succeeded in equating the problem of magnetization by influence of perfectly soft bodies by admitting the existence of magnetic particles and by supposing, moreover, that, for isotropic bodies, these particles have a spherical shape. But, for crystallized bodies, these particles may have a shape different from that of the sphere, and different effects will result. The passage in which Poisson foresees these effects and indicates the way to discover them deserves to be quoted. D. - II. i9 ago BOOK X. - MAGNETIZATION OF CRYSTALS. "The ratio between the sum of the magnetic elements and the whole volume in each magnetized body, says Poisson (*), is not the only data relative to this body, independently of its shape or its dimensions, on which the intensity of the magnetic actions can depend; the shape of the elements will also be able to influence this intensity, and this influence will have the particularity that it will not be the same in different directions. Let us suppose, for example, that the magnetic elements are ellipsoids whose axes have the same direction throughout the extension of the same body, and that this body is a sphere magnetized by influence in which the coercive force is zero; the attractions or repulsions that it will exert outside will be different in the direction of the axes of its elements and in any other direction; so that, if this sphere is made to rotate on itself, its action on the same point will change in general in magnitude and in direction But if the magnetic elements are spheres of equal or unequal diameters, or if they deviate from the spherical form, but are arranged without any regularity in the interior of a body magnetized by influence, their forms will no longer influence the results, which will depend only on the sum of their volumes, compared to the whole volume of this body, and which will then be "the same in every direction. This last case is that of wrought iron, and undoubtedly also of other non-crystallized bodies in which magnetism has been observed; but it would be curious to investigate whether the first case does not occur when the substances are crystallized. One could be sure of this by experiment, either by approaching a crystal to a freely suspended magnetized needle, or by making small needles cut from crystals oscillate in all sorts of directions and subjected to the action of a very strong magnet. " The problem of the magnetic induction of a crystallized body will thus be treated in the same way, according to Poisson, as the problem of the magnetic induction of an amorphous body; but, whereas, for the latter, one could attribute the spherical form to the magnetic elements, for the former, one will have to leave what (' ) Poisson, Mémoire sur la théorie du magnétisme (Memoirs of the Academy of Sciences, 1821 and 1822, vol. V). / 1 d\a = 1 in,= I - CHAP. I. - MAGNETIC EQUILIBRIUM ON CRYSTALS. 29! The shape and orientation of the magnetic elements is determined by the shape and orientation of the magnetic elements, assuming only that the shape and orientation are the same for all the elements. We find, according to Poisson, that at a point (x,y,z) of a crystallized body, the components Jl>, olh, G of the magnetization are related to the partial derivatives of the magnetic potential function by the equations (■) P> P'i P "i 9i ^'> ^"" ''5 ^'' 1 ''' being nine constants which, in the Poisson idea, are essentially positive. Poisson had indicated in advance the way by which one would succeed in discovering the phenomena of magnetic induction of crystals and the principles which would be used to explain these experiments. It was only twenty-three years later that the experimenters discovered the phenomena of which they had been informed. In 1847, Plûcker ( ^ ), having placed variously cut crystals between the two poles of a powerful electromagnet, observed the positions that these crystals took and thought he could conclude that the action of magnetism on the optical axes was different from magnetism and diamagnetism. Here is the experimental law by which he summarized his research: "If any uniaxial crystal is placed between the two poles of a magnet, the axis is repelled by each of the two poles. The force which produces this repulsion is independent of the magnetic or diamagnetic property of the mass of the crystal; it varies more slowly with the distance to the poles of the magnet than do the magnetic or diamagnetic forces from the same poles acting on the crystal. " (') J. Plùcker, Ueber die Abstossung der optischen Axen der Krystalle durcfi die Foie der Magnete ( Poggendorff's Annalen der Physik und Chemie, t. LXXII, p. 3i5; 1847). 292 BOOK X. - MAGNETIZATION OF CRYSTALS. In 1848, Faraday (*), having observed the orientation of a crystallized substance between the poles of an electromagnet, incidentally indicated that these phenomena could be explained "by supposing that the crystal was a little more apt to magnetic induction, or a little less apt to diamagnetic induction in the direction of the magnetocrystalline axis than in any other direction".) This explanation, to which Faraday did not stop, was that of Poisson. Faraday's work contained results in disagreement with those of Pliicker. The latter, resuming the study of the phenomena he had discovered, published two Memoirs (2) in which, without renouncing his first interpretation of these phenomena, he slightly modified the law he had thought to demonstrate in his first Memoir, by admitting that the action of a magnet pole on an optical axis was repulsive or attractive depending on whether the crystal was negative or positive. It is only after all these works, pursued in an inexact way, that Plûcker (3) on the one hand, Rnoblauch and Tyndall ('') on the other hand, came to recognize the true cause of the observed phenomena, and to attribute these phenomena, as Faraday had indicated, to a capacity of magnetization of the crystallized bodies variable according to the direction. "All these phenomena I have observed," says Pliicker, "can be explained by assuming that a magnetic or diamagnetic polarity (depending on whether the substance is magnetic or diamagnetic) can be (' ) Faraday, On the crystalline polarity of bismuth and other bodies, and on its relation to the magnetic form of force {Experimental Researches on Electricity, series XXII, art. 2588; Philosophical Transactions, pp. 1-41; 1849). ( = ) J. Plucker, Ueber die neiie Wirkung des Magnets auf einige Krystalle, die eine vorherrschende Spaltungsflàche besitzen. Einjluss des Magnetismus auf Krystallbildung {Poggendorff's Annalen, t. LXXVI, p. 676; iS^g). J. Plucker, Ueber die magnetische Beziehung der positiven und negativen optischen Axen der Krystalle {PoggendorJ/'s Annalen, t. LXXVII, p. 44?; 1849). (') J. Plucker, Ueber die Fessel'sche Wellenmaschine , den neuen Boutigny'schen Versuch und das Ergebnii^s fortgesetzter Beobachtungen in betreff des Verhaltens krystallisirter Substanzen gegen den Magnetismus (Poggendorff's Annalen der Physik und Chemie, t. LXXVIII, p. 421; 1849). (*) Knoblauch and Tyndall, Ueber das Verhalten krystallisirten Kôrper zwischen den Polen eines Magnetes {Poggendorff's Annalen, t. LXXIX, p. 233: i85o). CHAP. I. - MAGNETIC BALANCE ON CRYSTALS. It develops by induction in the crystals and develops there more or less easily according to the various directions, a fact which is linked to the variations of the elasticity of the ether. " Knoblauch and Tjndall concluded their Memoir thus: "Plucker's law, which attributes to the optical axes the particular way in which crystals behave between the poles of a magnet, cannot be preserved in its primitive form. "All the phenomena exhibited by Icelandic spar can be explained by assuming that magnetic samples are more weakly magnetic in the cleavage direction, and diamagnetic samples are more weakly diamagnetic in the same direction. )> The subsequent works of Faradaj, Plucker and Béer, Knoblauch and Tyndall, Hankel, and Matteucci constantly confirmed these conclusions. The theory, given by Poisson, of the magnetism of crystals had been forgotten by the authors of these experimental investigations. Sir W. Thomson (' ) called the attention of physicists to this theory. After recalling the ideas of Poisson and transcribing the equalities (i) to which the illustrious geometer had arrived, he added: "All the rest of Poisson's theory is limited to the consideration of the case of non-crystallized substances; in this case, it is shown that the coefficients/?, q\ r" are equal to each other and that the others are equal to zero. But this says nothing about the possibility of establishing general relations between the nine coefficients, whatever the nature of the substance. I have found that the following relations, by which these nine coefficients reduce to six, must be fulfilled, whatever the nature of the substance (2) j ^"=r, ( q' = p (') Sir w. Thomson, On the theory of magnetic induction in crystalline and non-crystalline substances {Philosophical Magazine, 4' series, t. I, p. 177; i85i. - Papers on Electrostatics, art. XXX). -294 BOOK X. - MAGNETIZATION OF CRYSTALS. "The demonstration of these relations is based not on an uncertain proposition or on a special hypothesis, but on the principle that a sphere of any substance, placed in a uniform magnetic field and capable of rotating about a fixed axis perpendicular to the lines of force, cannot become an inexhaustible source of mechanical effect. " This demonstration was later exposed by Sir W. Thomson (*). This remark of Sir W. Thomson completed the principles of a theory of the magnetic induction of crystals and established the fundamental equations. Also Sir W. Thomson (2) was able to combine the theory of magnetization by influence of amorphous and crystalline substances in a single study. Earlier Pliicker (^j had given interesting theorems on the properties of weakly magnetic crystals. A. Beer(*) had integrated the differential equations of the magnetic induction problem for a crystal sphere placed in a uniform chamj). Unfortunately, in the development of the induction-crystalline theory, he committed some analytical inaccuracies, as pointed out by E. Mathieu. E. Mathieu (^) arrived at equilibrium equations similar to those deduced from Poisson's theory; but he did so by a way that we cannot briefly analyze here. Let us only note that it is linked to a hypothesis (') Sir W. Thomson, Demonstration of a proposition on the magnetic induction of crystals { Papers on electrostatics, art. XXX, 1872). (^) Sir W. Thomson, General problem of magnetic induction {Papers on electrostatics, art. XL, 1872). (') J. Plucker, Ueber die Théorie des Diamagnetismus, die Erklàrung des Ueberganges magnetischen Verhaltens in diamagnetisches und mathematische Begriindung der bei Krystallen beobachteten Erscheinungen {Poggendorff's Annalen, t. LXXXVI, p. i; i852). - On the magnetic induction of crystals {Philosophical Transactions, i858, p. 543), {*) Béer, Einleitung in die Elektrostatik, die Lehre vom Magnetismus und die Elektrodynamik^ Brunswick, i865. (') E. Mathieu, Théorie du potentiel et ses applications à l'Électrostatique et au Magnétisme. 1" Part: Applications; Paris, 1886. CHAP. I. - MAGNETIC BALANCE ON CRYSTALS. agS on the way in which the magnetic actions are exerted within a crystallized substance and that it leads to a theory which, according to the admission of its author, cannot subsist if the magnets which act on the influenced crystal are not formed of the same substance as the crystal. We shall see that the principles laid down in Book IXj Chapter 1, easily provide the equation of the problem of magnetization by influence of a crystalline substance. § 2 - Magnetic induction surface. Let's consider a system that, for simplicity, we will assume is not electrified. This system contains a permanent magnet, which we will designate by the index 1, and a magnetized crystal, which we will designate by the index 2. The internal thermodynamic potential of this system has the following expression [Book IX, Chap. I, equality (19)] (3) 3i, J3, C being the components of the magnetization at a point along three rectangular directions O^, Or,, O^, parallel to the axes of elasticity of the substance at that point or to three directions invariably related to those. According to equality (20) of the same Chapter, each of the two functions ^J{3^, M, C) is of the following form: ( Ç(5l, B, €) = X5^ -t- [xB + (^C (4) +cpn(5V, B, C)5V2+cp22(5l, jD, C)iî2 + (P33(5V,lî, C)(!l2 ( -4-2cp23(5l, B, (!l)B "r4- 2cp3i(5l, B, C)C5V4-2cpi2(iA, B, C)5lB. The quantities \, [ji, v depend only on the parameters a, [3, ... which fix, at each point, the nature of the substance; the functions cpy,^ depend not only on ^, IP, C, but also on these parameters a, p, .... These functions remain finite for 31 = 0, B = o, C = o. 296 BOOK X. - MAGNETIZATION OF CRYSTALS. We will often have to consider the surface (5) j +2Cf23(2t,l3, C)Tl^-h2Cp3i(5V,j0,C)C^ + 2Ç,2(5V,B,C)?7) This surface of the second order, which depends on the nature and the magnetization of the substance at the point considered, is what we will call the surface of magnetic induction at this point. The knowledge of this surface, for any magnitude and any direction of magnetization, is the necessary and sufficient data to determine the magnetic properties of a medium. When the considered substance is holomorphic, we have X=0, [JL=0, v=o and equation (5) represents the magnetization surface referred to three axes passing through its center. § 3 - Existence of a state of equilibrium. The magnetization on the perfectly soft crystal 2 will be an equilibrium magnetization if it causes the quantity § defined by equality (4) to take a minimum value. In some cases, it can be predicted that there is a minimum for the quantity §, and, consequently, a stable equilibrium state for the system; the cases in question are those where the following conditions are verified: 1° The magnetic induction surface is, for any magnetization of the crystal 2, a real ellipsoid. 2° The substance which forms the crystal 2 is holomorphic. Indeed, consider the expression of the internal thermodynamic potential given by the equality (4) - The variations that we can make undergo to the magnetization of the crystal 2 cannot make vary the quantities E(r-TS) and A/i(5V, U, et, a, p, ...)rfPi. The quantity ^ can never be negative, and the same is true, according to the hypotheses made, of the" quantity J(^M^i î"" <^> "> P> ■-■)dVi. CHAP. I. - MAGNETIC EQUILIBRIUM ON CRYSTALS. 297 The set of values that the quantity § can take for any possible magnetization of body 2 is therefore limited below. Although it is not possible to conclude with any rigor the existence of a minimum for the quantity §, we can, at least, foresee this existence, which corresponds to a stable equilibrium state of the system. § 4 - Equations of magnetic equilibrium. Suppose that, inside the element dv^ of crystal 2, the components %, jw, C of the magnetization siiivant OÇ, Orj, O^ undergo arbitrary variations S 31, 815, ù^. If equilibrium is established on the system, the variation undergone by the quantity § will be equal to o. The components X, ilî), 3 of the magnetization along O^, Oy, O^ undergo variations Sjlo, 8i)l), §3. If '. dp -h [i + 'c^"*^t^'>t^^t)^ (6) + {^^"^^1w-^t^*t> + h^'^-t-^t^-ty-"' dp -f-V -H {^^"-^t-''ië--ii> -t K-*^-"^--'^')" 1 + s--^^-''^-^'^)^='" Poson S 4'!.= (^^■-'-*-s^-"t-'-^'s')' (7) < 4^27 = (,,"^5.^^,^x.^c<|i), h'/=^ (.,.,.^^|î^B^|.^c^-g-î). (8) (9) (^ = 1, 2,3); , dp tt = - ^ O31 -rp + 032 3- -t- O33 (n) 19 =-4^ ^.,^+822^+823 T V " c)^ <^^ (^C Let us assume that the system does not contain permanent magnets. Two cases will be distinguished: 1° The crystal is holomorphic. - In this case, }^, [jl, v being equal to o, it is easy to see that the equilibrium equations (6) or (11) are verified by posing 5\ = o, 13=0, C = o, because, if there are no permanent magnets, these assumptions lead to dp dp dp ^^^' d^=°' ^=^ Thus a perfectly soft holomorphic crystal will demagnetize if we remove it from the action of any permanent magnet. 2" The crystal is hemimorphic. - In this case, the quantities X, jjL, V are not necessarily equal to o. If at least one of these quantities differs from o, the equations (6) can no longer be satisfied if we make 5^ = 0, 13 =0, € = 0. Thus, in general, a hemimorphic crystal remains magnetized if we subtract it from the action of any permanent magnet. The experiment has never found so far on the hemi 3oO BOOK X. - MAGNETIZATION OF CRYSTALS. morphs ('), such as tourmaline, calamine, etc., this magnetism subsisting naturally, in the absence of any permanent magnet, the possibility of which is foreseen by theory. It must be concluded that, if the quantities X, pi, v are not equal to o for the known hemimorphic crystals, they have, at least, very small values. According to this remark, it would be useless, in the study of magnetism, to keep these coefficients "k, [*■■> ^] however, we will leave them in our formulas, because these same formulas will be of use to us again in the study of another problem where the coefficients "k, jx, v will play an important role. The equations (i t) show us that, in a holomorphic crystallized body, the direction of the magnetization no longer coincides, as in an isotropic body, with the geometric quantity whose ^ dp dp dp components are -y^ -p > -r^ § 5 - Determination of the distribution that is suitable for equilibrium. Let us suppose that experience has revealed, for a given body, of homogeneous or continuously variable structure, the magnetic induction surface represented by equality (5). Equations 7), (8) and (p) will then make T and the 8^^ a function of %, M, C and the x, y, z coordinates of the point to which these quantities relate. The quantities 1, [x, v can also be expressed in terms of x, y^ z. The relations (10) will become relations between %, 15, C, -r^-" -j-i - > x^ y^ z. We can suppose them solved in ^, jP, C, and write -^= 61 -3^' -T' "TÏF' ^' JK, -s \ o^ OTf aÇ, (12) "^^^^H"^'^'^'^'^'^ ^ . /dp dp dp (") É. Mathieu (Théorie du potentiel, t. Il, p. 166) was the first to indicate that hemimorphic crystals could, from the point of view of magnetism, be distinguished from hoiomorphic crystals. r CHAP. I. - MAGNETIC BALANCE ON CRYSTALS. 3oi We see then that, by means of these equations, where the functions 9< , O2, O3 are functions whose form is determined when the shape of the magnetic induction surface of the crystal is assumed to be known, the determination of the distribution that is suitable for equilibrium comes down to the determination of the function \'^. Let us see how the latter is determined. If, in the expressions of T and ùpq, we replace %, 15, C by their expressions (12), T and S^^ will become " . d-Ç dO dp functions of -.^-, -r- > -5 ^-y, ^, and we can pose, in general The equations (10) will then become (14) i3 = D" (^|.^X) + D,, (^--^ I-) ^ D. We will impose a last transformation on these equations. ^, 13, C are the components of the magnetization along three axes O^, Ot), O^, parallel to the axes of elasticity of the crystal at the point {x,y, z). Let be, at this point, rt, a', a" the cosines of the angles of Ox with O^, Or,, O^; b, b'j b" the cosines of the angles of Oy with O^, Or,, O^; c, c', c" the cosines of the angles of O^ with O^, Or,, OÇ. These nine cosines are functions of x, y, z, determined when we know the nature and orientation of the substance at each point. Let -A,, ill), G be the components along Ox, Oy, O:; of the magnetization at the point {x,y, z). We will have !5V ^ a oA.> 4- b Db -+- c 3, j3 = aM, -I- 6' m, -t- c' a, 302 BOOK X. - MAGNETIZATION OF CRYSTALS. We will have, on the other hand, (i6) dx dy dz d-Ç ,d-Ç ,,d-Ç ^d'Ç ar^ dx dy dz à-Q "d-Ç , "t)X? "d-Ç OL, dx dy dz These equalities (i6) allow to transform the quantity ■ d-Ç d-Ç d-Ç ^"'^^-df' 'dy d^' ^'^' e ,- j to-Ç d-Ç as a function of::^, ^-, x^ y, z, dx dy dz ' The equalities (i4), (i5), (i6) then give + (Cû?il -H c'di2 -H c" di3) ( 1- V \ dz a'X-hb'^ + c' e= (aâ?2i-(-a'c?22-(-""^23) f- + X -+- ( 6 .ijl + 6' fl?22 + 6" C?23 ) ( ^ -f- ,a ( c"^21 -H c' C?22 H- c" "?23 ) ("^ a'M, + è "e + c "8 = (a ^3, + "'^32+ a" ^33) ( v^ + X -+-(bd,i + b'd,,-^b "d,,) (^+1^ + ( c ^31 + c' fi?32 H- c" (3^33 ) /^ -p H- V or, denoting by (D"^ functions of t^, t- . ^,^, r, ;, MAGNETIC BALANCE ON CRYSTALS. 3o3 easy to form according to the dpq, ^) + CD,: ^) + (©2! /dp ^) + ©3; /d-Ç It now becomes easy to form the partial differential equation that the function "Ç verifies in all regions of space. Inside the permanent magnets 1 , we have and the second member is a known function of x, y, z. In the space between the permanent magnets and the crystallized body, we have (19) \-Ç = o. To obtain the partial differential equation that t? verifies at all points of body 2, let's difTerentiate the first of the equalities (17) with respect to :r, the second with respect to y, the third with respect to z, and let's add member by member the results obtained. The second member of the equality we arrive at will be a function of.r, y, z and the first and second order partial derivatives of the function "Q. The first member can be replaced by We obtain a second order partial differential equation that the function "C" must verify at any point of the crystal. To complete the determination of the function "Ç, it will be necessary to impose boundary conditions on this function. At infinity, or on the separation surface of the permanent magnets and the non dx 471 + dQ, dz 304 BOOK X. - MAGNETIZATION OF CRYSTALS. magnetic medium, these conditions will be the same as if the perfectly soft body were isotropic. At the separation surface of the perfectly soft body and the non-magnetic medium, we have ^. + ^ = 47: [-A cos(Ni, x) -f- Db cos(N:,^) + e cos(N/, z)], or else, by virtue of the equalities (17), 7^(|^ + ^)-[(©u'^os(N,-,a.) + ®2iCOs(N,-,7) + ®3iCos(N,,^)](g+x) (■-40) / - [(Di2Cos(N,-,a:)-i-(D22Cos(N,-,jK) + (D32Cos(N/,5)] f- +[xj - [(Qn cos (N, , x) -f- (D23Cos(^N,-,7) 4- (D33 cos(N,-, ^)] ( -^ + ^ ) =^ " - Thus is defined the problem that one would have in general to integrate to determine "Ç and, consequently, according to the equalities (12), the distribution which is appropriate to the equilibrium. § 6 Remark on homogeneous hemimorphic bodies. In the foregoing, we have not assumed anything about the homogeneity or heterogeneity of the body studied. In the case where the substance which forms this body has, at all points, the same constitution and the same orientation, we can join to what precedes an important remark. The axes of elasticity O;, Ot), O^ having the same orientation at any point (x, y, z), we can suppose that we have taken the axes Ox, OjK, O^ parallel to the axes of elasticity. The equalities (10) can then be written x = Di, = 3 ^ -4h(g-)-^K^ Moreover, if the body is homogeneous, "X, jjl, v are independent of X, y^ z. It is then easy to see that a homogeneous hemimorphic body magnetizes in a given field as if it were CHAP. I. - MAGNETIC BALANCE ON CRYSTALS. 3o5 holomorphic field and superimpose on the given field a unified field whose magnetic potential function would be \x -\- [i.y -\-v z. This proposition brings back, in the case of homogeneous crystals, the general study of hemimorphic crystals to the more particular study of holomorphic crystals. D. -II. 3o6 BOOK X. - MAGNETIZATION OF CRYSTALS. CHAPTER I THE FISH THEORY. § 1 - The two magnetic induction surfaces. We have seen that the study of the magnetization by influence of a crystalline mass supposes the knowledge of the magnetic induction surface defined by the equality 1^ -+- JXTj -f- V^ This surface is related to three axes O^, Ot), O^, whose directions indicate the orientation of the substance at the point (.r,jKj ^o they are, for example, the axes of elasticity at the point (x,y,z), or three rectangular directions invariably linked to the axes of elasticity. Jjappi'oximation of Poisson consists in supposing that the quantities '-fpq are, not functions of %, |5, C, but simple constants. The magnetic induction surface at a point no longer depends on the magnetization at that point, but only on the nature at that point of the substance of which the crystal is formed. Let us suppose that the crystal has the same constitution at each point; the surface of magnetization will be, for each point, the same surface of the second order. We have assumed so far that this surface is related for each point {x^y^ z) to any three rectangular directions O^, Orj, 0(^ invariably related to the axes of elasticity at the point {x,y, z). CHAP. II. - THE THEORY OF FISH. '307 But we can choose these lines in such a way that the terms in tX, ^i, iv] disappear from the equation of the magnetic induction surface. The three rectangular lines O^, Oyi, O^, which must be chosen for this purpose, are what we will call the axes of magnetic induction at the point (x, j , z). The equation of the magnetic induction surface related to the magnetic induction axes will be of the form (I) J._ + Jl_+ A_ 4.X^_l_f^r, +vC-l. aCTi 2T!T2 2T!73 "= l ' If the crystal not only has the same constitution at each point, but if, in addition, it has neither macles nor penetration groupings, the axes of magnetic induction will be oriented in the same way at each point with respect to the axes Ox, Oy^ Oz. In any question where the position of the crystal in space does not vary, it can be assumed that the axes of magnetic induction of the crystal are fixed as Ox, O^, 0-s. For example, let's study the magnetization of the crystal by taking the magnetic induction axes Oç, Oyi, OÇ as coordinates. Let XD be the magnetic potential function of the permanent magnets 1 andfOa the potential function of the magnetization distributed on the crystal 2. The variable part of the internal thermodynamic potential has the value 1 J W ^Ç2 2 J II dç, + / - -{ -{ ^ 4-X5V2+^i32-+- v€2)0?^'2- l J \2nJi 21^2 2TO3 ^ Equating to o the variation that this quantity ^ experiences as a result of any change in magnetization, we find the equalities a ^ _ ^, ^A ^ _ ^_ _ (3) ^fi=_^,(^^ + __ + _ 3o8 BOOK X. - MAGNETIZATION OF CRYSTALS. These relations show us how the geometrical quantity must be directed which has for components - [x-t dXDi -+ atXDA --%, + dtl ) -h dXDi -4 = Y> so that the magnetization has a given direction. The rule that fixes the direction of the quantity a, [3, y is as follows: Through the center Q {fig' ^2) ^^ ^^ magnetic induction surface we lead a straight line QOIL having director cosines propor Fig. 22. W, C Through the point where we lead a plane P conjugate, in the magnetic induction surface, to the direction QOII. Finally to this plane P and on the same side as the line QOll/, we lead a normal QF. The director cosines of this line QF are proportional to a, p, y. When the direction of the geometrical quantity with components a, ^, y is given and we want to obtain the direction of the magnetization, we can repeat a construction similar to the previous one by using not the magnetic induction surface, but the inverse magnetic induction surface, which has the equation (4) nTiç2_(_ nT.2rj2-i- 1573^2 = i. CHAP. II. - THE THEORY OF FISH. Sog From either of these constructs, the following conclusion is deduced: For the geometric quantity whose components are a, p, Y to coincide in direction with V magnetization, it is necessary and sufficient that these two geometric quantities are directed along one of the magnetization axes of the crystal. The three quantities gt, nT2, gts, which are related by very simple relations to the lengths of the axes of the direct and inverse surfaces of magnetic induction are called main coefficients of magnetization of the crystal, § 2 - The determination of the magnetization reduced to the integration of a partial differential equation. The magnetic potential function X!), of the permanent magnets is assumed to be given. From then on, according to the equalities (3), the determination of the components ^, w, C of the magnetization comes down to the determination of the function Do. Let's differentiate the first of the equalities (3) with respect to i, the second with respect to rj, the third with respect to J^, and let's add member by member the results obtained, observing that dHOa d'-'O'i ^HOn _ (d% d^ d€\ and that, in any point of the crystal, d^XDt ano, ()2X5, d^-^ dr,^ c^C'' we will have o; ,,. , I , \cJ2€>2 ,/ I , \d^t% , / I \d^^ (4) 7^ + ^1 -Jfir -^ T- + ^2 -3rv- ^ 7^ + ^3 This is the partial differential equation that the function tDj must verify at any point of the crystal, while at any point outside the crystal, it verifies the partial differential equation d^t)i d^XDi to^^i The function XD-i is equal to o at infinity as an ordinary potential function; it is continuous in all space, even at (6) 3 10 BOOK X. - MAGNETIZATION OF CRYSTALS. the crossing of the surface that separates the crystal from the surrounding medium. This is not the case for its partial derivatives of the first order. We have, indeed, at this surface, equation which becomes, by means of the equalities (3), L^.,)(:|3)_eos(N,,0+(^H-..)(^)^cos(N,,,) f'),.cos(N,,Ç) I 4" 1 4^ TïTs I aOî These various conditions cannot leave any indeterminacy in the expression of the quantity XD2- Let us suppose, in fact, that we can fulfill them by means of two distinct expressions of t)2 and be % the difference of these two functions Og. The function 0, it is easy to see, will verify an equation similar to equation (4) at any point inside the crystal and, outside, it will be harmonic. If du designates an element of the unlimited space outside the crystal, we must have -^/[(4Î^ -^ "0 9 -^ (4-^- "- "0 ^^" (4^ -^ "^) ?] ^ ^'^^ ^ " If dS denotes an element of the surface which limits the crystal, an integration by parts will transform this equality into Uli-!: i^ dNe I 4tc But, the two expressions of the function Dg verifying the equality (6) THE FISH THEORY. 3ll at any point of the surface S, we have, at any point of the surface S 4uc^N. ■ (4i;-^"0(l).-'''^'^''^^"^fe-^"0(^).'"'^'^^ 4tî'^^7 \dri)i The previous equality will thus be reduced to We will see later that it is only necessary to study crystals for which the three quantities ^i, Tn.21 ^3 are positive. The previous equality cannot then take place if we do not have, at any point in space, a positive value for the three quantities ^i, Tn.21 ^3, dS d9 of If we add that the function is continuous in all space and equal to o at infinity, we see that we must have, in all space, = 0, so that the two expressions of the function XD.2 are identical. Thus, when the three principal coefficients of magnetization are positive, there cannot be more than one solution to the problem of V magnetization by influence. Moreover, what was said in §§ 3 and 6 of the preceding chapter leads us to predict that, in this case, this problem always admits a solution. § 3 - Stability of the magnetic equilibrium. Does the magnetic distribution determined by the above equations correspond to a steady state? Let us consider an element of volume dv^ belonging to the crystal, and let us give twice in succession to the components^, ÎB, C of the magnetization in this element arbitrary variations 85t, 8Î5, 8C. The first variations make 5* grow by a quantity 5cf = -~ -f- ---^ -H À H 1 ôJV 0% aÇ T "Tj 3li BOOK X. - MAGNETIZATION OF CRYSTALS. The second variations vary 8é^ of 82#: (851)2 (8îî)2 (S(jr)2 CTi THj If the three quantities r;3<, nr^î ^3 are not all three positive, it will always be possible to choose the quantities ù%, ùM, 0^^ so that the quantity 8^ éf is negative. Thus /'e^"/i7/6/-e magnetic can be stable on a crystal only if the three main coefficients of^ magnetization are positive. Conversely, if the three main coefficients of magnetization of a crystal are positive, the magnetic equilibrium is certainly stable on this crystal. Let's give twice in a row to the magnetization at each point of the crystal arbitrary variations S^, SU, 8C Let's put ^\ = a dt, oj(3 = 6 dt. oC = c dt, dt being an infinitely small constant and a, b, c being finite continuous uniform functions of ^, r,, "C^. Let us suppose that at the various points of the crystal we distribute a magnetization with the quantities a, b, c as components. Let Q be the magnetic potential function of this magnetization. By reproducing the calculations made in Book IX, Chapter III, § 2, we find without difficulty that we have If, as we have assumed, the three quantities ctj, nT2, W3 are positive, this quantity S^J^ is essentially positive, and the magnetic equilibrium is stable. Thus, in order to determine on a crystal, placed under given conditions, a magnetic distribution corresponding to a stable equilibrium, it is necessary and sufficient that the magnetic induction surfaces are real ellipsoids. The crystal cannot be diamagnetic in any direction. CHAP. m. - CRYSTAL IN A UNIFORM FIELD. 3l3 CHAPTER III. ACTION OF A UNIFORM MAGNETIC FIELD ON A CRYSTALLIZED BODY. g 1. - Magnetization of a crystalline sphere or ellipsoid in a uniform magnetic field. A. Béer (') showed that the method by which Poisson studied the magnetization of an isotropic sphere placed in a uniform magnetic field could easily be extended to the case where the sphere is cut in a homogeneous crystalline substance. To solve this problem, we can take the coordinate axes parallel to the principal axes of magnetization of the substance and led by the center of the sphere. The equations of the magnetic equilibrium will then be [Chap. JI, equalities (3)], 3V = - You ( X -h U = - TÎI2 ( |X + à', -) + + this Let's suppose that the field, uniform, corresponds to a magnetic potential function t!), = _(F$-f-Gvi + HÇ-i-K), (') A. Béer, Einleitung in die Elektrostatik, die Lehre von Magnetismus und die Elektrodynamik. Brunswick, i863. 3l4 BOOK X. - MAGNETIZATION OF CRYSTALS. F, G, H, K being four constants. The previous equalities will become (0 |5=-n.,(fx-G+^) These equalities unambiguously determine ^, M, C; and it is easy to see that they will be verified by a uniform magnetic distribution spread inside the sphere, and having in each point as components, along the three main axes of magnetization (2) 4 I -i- ^ TTHJi 7^2 4 I -4- - TTIHa 7^3 4 1 ■+- - TTTÎTs €= 7 (H-v). In this case, indeed, we will have [Book VIII, Ghap. II, equality (5)] t)2(?,iri,!:)=^7:(5V$+î3Tl + CO, at any point (Ç, r, ^) inside the crystal sphere. An analogous method will be applied to the study of the magnetization taken, in a uniform magnetic field, by an ellipsoid cut in a homogeneous crystal. Let us take as coordinate axes the principal axes of the ellipsoid, Ox, Oy, Oz, which do not necessarily coincide with the principal axes of magnetization O^, Oy], O^ of the crystallized substance. The nine cosines of the angles that the lines of the first trihedron make with the lines of the second trihedron will be denoted according to the following table: CHAP. III. - CRYSTAL IN A UNIFORM CHAUP. 0$ Ot) OÇ Ox 3t5 O7 Bones a a' a" b b' b" c c' c" Therefore, if we denote by .A.,, i)l), 3 the components of the magnetization along O^, Oy, Os, we will have .A = 5la -1- j3a'H- €a", G = 51 c -+- 13 c' -i- C c", or, by virtue of the relations (i), oA =- [nj,(X - F)a -4-nT2([JL- G) "'m- T33(v - H)a"] nri - a + TîT^^a+nTs- a (3) Dî, = - [r;ji(X - F) 6 + nT2( [JL - G) 6'-+- TOs (v - H) 6"] -.(r.,-^b-^m,-^b+r^,-^b), G = - [mi (X - F) c + nT2 ([J. - G) c'-}- W3(v - H) c"] dXDo ^3 Let's see if it is possible to satisfy these equations by filling the ellipsoid with a uniform magnetic distribution. If the ellipsoid is uniformly magnetized, we will have [Book VIII, Chap. II, equality (10)], dx = 2X/, dy alJbm, dz = ikin, designating by /, m, n constants that we had designated by X, [jL, V in Book VIII, Chapter II. 3l6 BOOK X. - MAGNETIZATION OF CRYSTALS. These last equalities give - - = 2 ( Jlg m ^^~ = 2 {Xla'-h 1)1) m^' H- Bnc'), 1 ( JU la -+- Dî, mh" -f- G ne"). ^2 dV It will therefore suffice to determine X, i)b, G by the following three linear equations [i + aZ ("uia^ -+-vj.2a'^ -i-rnsa"^)] JU -I- 2/n(nji "6 -I- rs^a' b' -+- xn^a" b" ) ift) -f- 2n (nji ac + TJÎ2 a' c'-f- T "j3a "c") S = cTi a ( F - X ) 4- 7IT2 a' ( G - [j. ) -;- CT3 a" ( H - v), aZ (xni 6a-î- TOj 6'a'-i- 75T3è "a")(^l) -t- [ I + 2 m ( HTi 62 -f- TÎT, è'2 -H 77T3 //'2 ) ] 1)1, t- 2/1 (tTTi èc - T!Ti 6'c' + Tx!ib "c") G = TjTi 6 ( F X ) H- 7TT2 6' ( G - [Jl ) + Tn3 6" ( H - V ) , (4) 2 Z ( TJTi Ca + 7^2 c' a' -i- TO3 c" a" ) aH" M- 2 7n(T0i c6 -f- nT2 c' b' -T- nj3 c" 6 " ) 1)1) -l-[t-+-2" (tîTi C^ -!-7n2C'2 -t-7n3C "2)] G = TiTic(F - X) + nj2c'(G - [x) -~- m-ic" {H - v ). Thus, an ellipsoid cut out of a crystalline substance and placed in a uniform magnetic field magnetizes uniformly. § 2 - Forces acting on a crystalline sphere in a uniform magnetic field. A homogeneous crystallized body being placed in any magnetic field, the internal thermodynamic potential of the system, reduced to those of its terms which are likely to vary, is given by the equality (2) of the preceding chapter; this equality can also be written 73, dvi J W \ to^2 c^^2 ^1 / 2j II d^z But, according to the conditions of magnetic equilibrium, we have, at CIIAP. m. - CRYSTAL IN A UNIFORM FIELD. every moment, ^2 = ^1 I A + -rz H -TTT- 1 ) c?r)2 dt\^ ] ' 3i7 CJ2 [Jl + The previous equality is therefore reduced to (5) 2j H t^? 512 C't'-2. This equality is true, whatever the particularities presented by the magnetic field and whatever the shape of the crystallized body. In particular, let us assume that the field is uniform and that the body is spherical. The sphere will magnetize uniformly and, at any point (^, Y), X^ inside the sphere, we will have t)2(?,ri,C)-f 7r(5l^ + iDri ■-Q The equality (5) becomes therefore, by designating by R the radius of the sphere. .f = -3^ TTTÏTi -5V2 R3, or, by replacing ^, W, C by their values (2), (6) i= [4 4 n-3^Tn, (F-X)2 4 l 4- - 7rTJT2 -(G-[Jt)2. 7 (H-v)2 I+.-7rCT3 J If the orientation of the sphere varies in space, the F, G, H components of the field action along the main axes of magnetic induction of the crystal will undergo variations, and the quantity i^ given by the previous equality, will undergo a variation '^i. The quantity ( - 8^) will be the work done by the magnetic forces to which the sphere is subjected. A virtual translation, unaccompanied by rotation, does not vary the quantity $. The magnetic actions that a 3l8 BOOK X. - MAGNETIZATION OF CRYSTALS. Any crystalline sphere in a uniform magnetic field can therefore be reduced to a couple. Expression (6) is susceptible to a geometric interpretation. Consider the ellipsoid (E') represented by the equation (7) This ellipsoid has the same axis directions as the magnetic induction ellipsoid. If, through the center O, we lead the geometric quantity whose components are F-X, G - IX, H~v, its direction will meet this ellipsoid at a point M'. Let p' be the distance OM'. Let P' be the tangent plane at M' to the ellipsoid E'. Let A' be the distance from the center O to this plane P'. We have l^rr., - irnj2 ô ^^3 -. $- + I H- i Tïnji 4 ^''-^ 4 1 -+- - TTTOi I -h - TITHs 4 4 - TTTIJi - TrT 7372 = 7^3 , CIIAP. III. - CRYSTAL IN A UNIFORM FIELD. 325 and the torque will have the value ■ ^ 4 3^nT, 4 1 + - TTTIT, 4 R2J2 j _ jcosîçrf'];. The duration of oscillation of the crystal slightly deviated from its equilibrium position will have a value T given by the formula C^) h = V^\^, ^)co.= ,. In particular, suppose that O^ coincides with O^. We will then have cp = 71, COSCp = - 1, and the duration of oscillation Tq will be given, according to equality (i3), by the formula J__ R2J2 So we have (II) This remarkable equality was verified by Plucker by means of a sphere cut in a crystal of iron sulfate. We have seen that this substance was magnetically uniaxial. § 4 -- Action of a uniform field on a weakly magnetic crystal immersed in a weakly magnetic medium. The various theorems which we have just proved do not assume anything about the magnitude of the principal magnetization coefficients of the crystal. We shall now set out the approximate propositions that can be established by assuming these coefficients to be very small. This theory, given by Plucker (') in i858, has two advantages: first, it can be established in sup (') Plccker, loc. cit. 4 - !-■ 4 4 I + - TrnT2 X2 -Hcos^ ?- 326 BOOK X. - MAGNETIZATION OF CRYSTALS. Secondly, it allows us to extend to crystals of any shape properties which, in the previous theory, were only presented by spherical bodies. These two advantages justify the development that we are going to give to the presentation of this theory. By reasoning similar to that used in Chapter VII of the preceding book, it will be easily proved that the weakly magnetic crystal and the weakly magnetic medium in which it is immersed both magnetize as if each of them existed alone. The equations of the magnetic equilibrium, related to the main axes of magnetization of the crystal, will be, for the crystal, ■--".('- 1)' (i5) { 132 = - T €2 = - THa and, for the environment, 5I3 = - A-3 (i6) ^^=-k3--!-, ("t3--A3-^ If the field is uniform, both the body and the medium magnetize uniformly. In the medium, the magnetization is directed as the field strength created by the permanent magnets. This is not the case in the crystal. By reproducing considerations similar to those we indicated in Chapter II, § 1, we arrive at the following result: In the inverse ellipsoid of magnetic induction one leads a half-line D whose directing cosines are proportional to -v^^r -\^-^-ô^r -y^-^)' CHAP. III. - CRYSTAL IN A UNIFORM FIELD. 827 through the center of V ellipsoid, we lead the plane conjugate to this direction D, and, on the side of this plane where the direction D is, we lead a normal N. This normal marks the direction of magnetization. The variable part of the internal thermodynamic potential of the system can be written as ^2 the volume of the crystal and ç^ the unlimited space occupied by the magnetic medium, J 11 ''^2 11 J 11 OZî i J J \2TCTi 2UT2 2X33 / We will have, moreover, by virtue of the equalities (i6), f[ li^' ^ Il"^'sC^r^a)] d^^z = -J^'zV^\U)dv3. Moreover, by virtue of a theorem proved in Book IX, Chapter IX, § 1, the sum /Va ( OÏL 3 ) dv^ ^ fws ( Oit 3 ) dv^ does not vary. Finally, according to a calculation made in § 3 of the same Chapter, we have fw3{DTi3)dv.2 =^ fnXDidi^i. If we keep the notations of the previous paragraphs, we will have and the internal thermodynamic potential, reduced to its variable terms, can be written r r/ it =* àVt \ 3ii m, €1 . ^ 3, " \ , s = I i \%i-^ -\ -i H -i- X?l2-f- u.i32-l-v€2) dVi J \\\ d^2 I 2nTi 27^2 2TiT3 / -t-^ f{F^^G^^H^)dv2. The equalities (i6) bring to this expression a last trans 328 BOOK X. - MAGNETIZATION OF CRYSTALS. training and allow to write (,7) .f =- 2[T7Ti(F-X)5-+-nT.2(G-|JL)Mra3(H-v)2-A:3(F2 + G24-H2)]Vj. When the crystal is moved, the quantity ^subjects a variation Srf equal, to the nearest sign, to the work of the magnetic forces acting on the crystal. The form (17) of the quantity ^ shows us, in the first place, that the magnetic actions exerted on the crystal are reducible to a couple. The expression (17) is susceptible, in the case (which is the only one interesting for the study of magnetism) where we have X = O, [JL = O, V = O, of a geometric interpretation similar to the one we have given for expression (6). In this case, in fact, equality (17) can be written (18) ^ = - '-[(T^,-/c,)¥^ + {v,,-k,)G^-^{m,-k,)H^]. Consider the two second order surfaces S and S' defined, the first by the equation (19) - --r -^ -r-\ -T- = i> HTi - A'3 nj2 - A-3 THs - K3 and the second by the equation (20) (nîi.-A-3)^'2 + (nT2-A-3)r/2 + (T03-/t3)r=i. These two surfaces are not necessarily real ellipsoids, as are the surfaces E and E'; they can also be imaginary ellipsoids, or hjperboloids with one or two layers. In all the cases that have been experimentally studied, the three quantities ^1 - "^3) ^2 - ""3i ^3 'î "3 have the same sign, so that, in all these cases, the surfaces S and S' are either two real ellipsoids, or two imaginary ellipsoids. In the first case, the crystal is said to be paramagnetic; in the second case, it is said to be diamagnetic. CHAP. III. - CKISTAL IN A UNIFORM FIELD. SîQ Let us lead, through the common center O of the two surfaces S and S', a half-line directed as the intensity of the field. It meets the first at a real or imaginary point M and the second at a real or imaginary point M'. Let's say OMr=p, OM'=p'. Let J be the field intensity. We will have (21) i = t'îJî p2 :i'2 ■ formula that it will be easy to discuss as we have already discussed equality (lo). Let us assume, first of all, that the crystal is absolutely free to orient itself. It will reach its stable equilibrium position when i will have taken its minimum value, i.e. when ^ will have taken its P maximum value. If we have TiTi > r;T2> TIT3, This position will be reached when the main axis of magnetization OÇ is parallel to the lines of force of the field. Let us assume, secondly, that the crystal is only free to rotate around one of the main axes of magnetization, the field strength being normal to this line around which the crystal is mobile. The other two axes will generally correspond to main magnetization coefficients of unequal magnitude. In this case, the one of the two moving axes that corresponds to the largest magnetization coefficient will be placed along the lines of force of the field. If the crystal is magnetically uniaxial and if the axis of revolution of the magnetic induction ellipsoid coincides with the axis around which the crystal can rotate, the equilibrium of the crystal will be indifferent in all positions. In general, the two surfaces S and S', defined by equalities (19) and (20), do not admit the same directions of circular sections. But, in practice, it happens that the planes of the circular sections of the surface S differ very little from the planes of the circular sections of the surface S'; just as, in Optics, the 33o BOOK X. - MAGNETIZATION OF CRYSTALS. The inner conical refraction axes differ little from the outer conical refraction axes. This being the case, let us make the crystal mobile around a straight line which differs little from the normal to the circular sections of the surface S and from the normal to the circular sections of the surface S' and let us suppose the lines of force of the field normal to this straight line around which the crystal can turn. The equilibrium of the crystal will be almost indifferent. ClIAP. IV. - LOW DEFORMED BODIES. 33 1 CHAPTER IV. MAGNETIZATION OF SLIGHTLY DEFORMED BODIES. § 1 - Magnetization of any body with little deformation. JWe have seen that raimanlalion of any perfectly soft body depends on the function (0 -f-cp"(5l,jB,€)5l2 +cp22(5l,ÎJ,C)ÎJ2 -4-033(-9l,îî, "D)r2 ( +cp23(5^,î9,C)î3C + cf3i(5l,ÎJ,C)€5V-f-cpi2(3i,j3,C)5VJ&, /3i, M, C being the components of the magnetization along three rectangular lines whose orientation is known when the nature of the body at the point under consideration is known; these may be, for example, the axes of elasticity of the substance at the point under consideration. The three coefficients [x, v and the six functions ^pq{^, H^, ^) change if the state of the substance at the point under consideration is modified, if, for example, it is deformed; hence the need to study especially the magnetization of a deformed body. The formulas which we will arrive at in this study will be of great use to us in the research which will be exposed in Books XI and XIL Let us imagine a body which, from a certain initial state, called natural state, has undergone a small deformation. Let us take any point M inside this body. Through this point let us lead the three rectangular lines M^, Mri, M^, which served as axes of elasticity to the matter which is in this point before the deformation. Let us assume that the body is homogeneous in its natural state, so that these three straight lines have the same direction at every point of the body. d\] dY dW ^V dri j ôC -+ dW -\ -h 332 BOOK X. - MAGNETIZATION OF CRYSTALS. The deformation in question has given to the point M, with respect to a fixed coordinate system, a displacement whose components along M^, Mt), MÇ are U, V, W. We know then that the very small deformation undergone by the substance which is at the point M will be defined by the knowledge of the six quantities (^) i, Tj, Ç being the coordinates of the point M, with respect to a coordinate system O^, Ot), O^ whose axes are parallel to Mi, Mri,MC. The three quantities )v, p., v and the six functions ©^^ which appear in the expression (i) of (j(5l, lî, C) will be functions of the six deformations (2); they may be regarded as linear functions, if the six deformations are all very small. We can, for example, write the seven quantities "ko, /,, lo, I3, //, /s, la depending on the nature of the substance in the natural state. It is easy to see how, from this starting point, one could deduce a complete theory of the magnetization of slightly deformed bodies. We do not intend to develop this theory in its entirety here. We shall limit ourselves to insisting on a few particularly important questions. The first of these questions is related to the changes that the quantities X, pi, v can undergo by the effect of a deformation. If the constitution of the substance in its natural state is holomorphic, a case in which a center is among the elements of symmetry of this substance in its natural state, it is easy to see that the weakly deformed substance is still substantially holomorphic. Thus, for a substance whose symmetry has a center, A is equal to o whatever the six deformations are. CHAP. IV. - LITTLE DEFORMED COaPS. 333 lions (2); in other words, we have Xo = 0, /i = O, /" = O, ^3 - O, l\ = 0, /s = O, /g - O. Let us now consider hemimorphic substances, that is, substances whose constitution does not admit of a center of symmetry. Although such a substance, in its natural state, does not admit of a center, it can happen that we have, for this substance, Xo = o, [^0 = o, Vo = o. Indeed, the system formed by the origin of the coordinates and any of the magnetization surfaces represented by the equation / -^ ;jLr, -f- v^ (3) ^oh(5V,?, C)Ï2 -4-cp22(5V,i3,£)r,2 + cp33(5V,î', G)C^ must always have a symmetry at least equal to that of the substance to which it refers. Now this condition can require that this surface always has for center the origin of the coordinates without that I crystal has a center. Let us imagine, for example, that the symmetry of the substance considered has an axis of any order, binary, ternary, quaternary or senary. A rotation of an angle equal to or less than tt about a line parallel to this axis, led by the origin of the coordinates, will have to bring back to its primitive position the surface (3). This requires; 1° That a parallel to the considered crystallographic axis, led by the origin of the coordinates, passes through the center of the surface (3) ; 2" Let this line be one of the main axes of the surface (3). If the substance admits another axis of symmetry, one can state for this second axis analogous propositions, and the center of the surface (3) must coincide with the origin of the coordinates. The same will be true if the substance admits a plane of symmetry normal to the first axis. 334 BOOK X. - MAGNETIZATION OF CRYSTALS. Thus, we can necessarily have for a substance (4) l - o, [J. = o, V = o, even if this substance does not have a center; it is sufficient that it has more than one axis of symmetry (\), or an axis of symmetry and a plane of symmetry normal to this axis. If we take, for example, the plagiarized quartz in its natural state, this body does not have a center; but it has an axis of ternary symmetry and a plane of symmetry normal to this axis; the three coefficients )., [x, V will be equal to o for the plagiarized quartz in its natural state The substances presenting the elements of symmetry which we have just considered are brought closer, one sees it, of the holomorphic substances by this fact that, for the ones as for the others, one has X - o, [J. = o, V = o. But a radical difference separates them. For a holomorphic substance, we have X = o, [J. = o, V - o, not only when the substance is in its natural state, but also when it has undergone a slight deformation; for it is easy to see that a substance which has, among its elements of symmetry, a center in its natural state, still has a center when it has undergone an infinitely small deformation. This is no longer the case for substances which, without admitting a center, admit several axes of symmetry, or an axis of symmetry and a plane of symmetry normal to this axis. For these substances, taken in their natural state, the equalities (4) are verified, but they may no longer be verified after an infinitely small deformation altering the primitive symmetry of the substance. An important remark concerning this case is the following: Let M {Jig. 24) i^in point taken inside a body which, in the natural state, has a plane of symmetry P; MN is the normal (' ) See Mallard, Traité de Cristallographie, t. II, p. 571 - According to this, the only bodies in which X, [x, v can differ from o are those to which the German crystallographers reserve the name 'hemimorphs'. CHAP. IV. BODIES WITH LITTLE DEFORMITY. 335 to this plane of symmetry. Let us subject the body to a slight deformation. The plane P becomes P', the direction MN becomes MN', the latter no longer being normal to the plane P'; it is easy to see that the plane P' is, for the deformed substance, at point M, a plane of oblique symmetry, the direction of sjmetrife being MN'. It is easy to conclude that the geometric quantity Çk^ Y-^^) '^'"^ P®^ ^^ component along MN'; it is normal to MN'. As, moreover, the direction MN' differs infinitesimally little from MN, we can also say that the quantity (X, [x, v) has no component along MN. These various remarks, not very useful in the study of magnetism where, up to now, the influence of the quantities X, p., v has not been noticed, have a preponderant importance in the study of pjroelectric and piezoelectric crystals, as will be seen in the following Book. § 2 - Magnetization of a slightly deformed isotropic body. Consider a solid body, isotropic in its natural state. Let us imagine that this body experiences a slight deformation of some kind. The state of this body at a point is defined when we know the orientation of the main axes of expansion OX, OY, OZ and the magnitudes /, /', /" of the three main expansions. Relative to these axes, the magnetization surface represented by equation (3) must be reduced to the form (?u(Â, B, G)X2-+-cp,2(A, B, G)Y2+cp33(A, B, C)Z2=i, 336 BOOK X. - MAGNETIZATION OF CRYSTALS. A, B, C being the components of the magnetization along the three main axes of expansion. The quantities (Çh (A, B, C), cooo (A, B, C), cpsj (A, B, C) are functions of /, /', /". If the quantities /, /', /" are very small, we can admit that they appear linearly in the functions in question. Some considerations of symmetry will show without difficulty that we must have cpii(A, B, C)= F(A, B, G) + Z G(A, B, G) + iU{X, B, G)^ /" H(A, G, B), ca22(A, B, G)=F(B, G, A)-+- /' G(B, G, A)+ rH(B, G, A)- / H(B, A, G), cp33(A,B,G)=F(G,A, B)+rG(G,A,B)-f-ZH(G, A, B)^-/'H(G,B, A). Let's assume that Ton has 1 = l'= /"■ the body, initially isotropic, will still be isotropic after the deformation. The magnetization surface will, in this case, be reduced to a sphere and, by designating by Dit the magnetization intensity, have an equation of the form [/(DrL)+3/^(DH)](X^-^Y2-.^Z2)^,. We must therefore have, whatever A, B, C, F(A, B, G)H- l[G{X, B, G) + H(A, B, G)^ H(A, G, B)] = /(Ole)-*- 3^^(011), F(B,G, A)^/[G(B, G,A)-f-H(B,G,A)4-H(B,A, G)]=/(^lI)-^3/^(Dll), F(G, A, B)-4- /[G(G, A, B)^H(G, A, B)^H(G, B, A)] = /(.m)-^3 /"-(OR). Making first /=o, we find F(A, B, G)=F(B, G, A)=F(G, A, B)-/(,m). 11 remaining G(A, B, G)-l-H(A, B, G)-t-H(A, G, B)=3ff(0\L), G(B, G, A)+H(B, G, A)-t-H(B, A, G)=3^(D1L), G(G, A, B)-^H(G, A, B)+ H(G, B, A)=3^(31L), These equalities can be satisfied if one poses H(A, B, G)=: n(A, G, B) = H(B, G, A).= H(B, A, G) = H(G, A, B)=H(G, B, A)=/i(01L.) and G(A, B, G)= G(B, G, A)= G(G, A, B)= h'iDK). CHAP. IV. - THE LITTLE DEFORMED BODIES. 337 If we then take k{3\L)=h'(3\L)—h{DÏL), the magnetization surface, referred to the main axes of expansion, will have the equation (5) ] -^[A3ïL)^(l -h r -h l")h(DXl)^ r k(3ïL)]Y^ ( +[/(31L) + (Z + r-^ r)h{DTL)-\- rA^(^L)]Z2 = r. Let us now look for the equation of this surface related to any rectangular axes O^, Ori, O^. Let U, V, W be the components along these axes of the displacement of the point to which our surface refers. Let us adopt the following notation for the cosines of the angles of the axes OÇ, Ov), O^ with the axes OX, OY, OZ. o^ Ot) oc OX OY OZ . ^i ai Cl Clî 62 ^2 "3 63 C3 We will have the following relations, proved by Cauchy, and which are frequently used in the theory of elasticity: i^i + i ____^_ + _, garlic ^ garlic' -^ garlic" =^-^, , bii'^bir = c\i-^c\i' + cl r = bicj -^ b-iC^l' ^biCsl" = - -- 2\dZ, I /dY dW\ àrj' c^aj -^ c-ia^l -^Cia^l' = - {'ZJÎ"^ Ir ) ' A / A /' /. 7. ï /^U ôY aibil -+- a^bil -h azbil - -( -r 2\07i dY\ D. - II. 338 BOOK X. - MAGNETIZATION OF CRYSTALS. and the equation (5) of the magnetization surface will become dU + ^ + -:ïf +^-(31^) à^ > k (-)^} ^(31^)-]^^ [/(01L)+A(0rt)(^ -[/(01L)+A(DlI.)(^i^ Once this equation of the magnetization surface is known, it becomes easy to study the magnetization of a slightly deformed isotropic substance. Let us deal only with the case that corresponds to V Poisson approximation, i.e. the case where we have /(01I.) = /, A(01L.)=A, k{d\-i)=k, f, h, k being three quantities independent of the intensity 3\L of the magnetization. We will then have [Ghap. I, equalities (6)] (7) ô\ d\\\ d\\\ , ^Ul ^ j /du dV /dW dU\^ dV dU k^^V<&+ik{ df] J /d\J dY 5l= r. j /dV dY dW\ ,dW\^ , /dW dl]\^ d^ dl (dY d\Y -, d-0 These are the equations that determine, in this case, the components %, %, C of the magnetization along the axes O^, Ori, OJ^. BOOK XI. DIELECTRIC BODIES. CHAPTER ONE. THERMODYNAMIC POTENTIAL OF A SYSTEM CONTAINING DIELECTRICS. g 1. - Thermodynamic potential of a system containing electrified and polarized dielectric bodies. After having exposed, in the preceding Books, the principal properties of magnets, we shall return to the study of electrical phenomena and examine the properties of dielectric coi'ps; the calculations which we had to make in exposing the theory of magnetized bodies will be able to be used almost in their entirety in the research of the properties of dielectrics. It is, moreover, by imitating the representations that had been adopted to summarize the phenomena presented by magnets that physicists also managed to represent the phenomena presented by dielectric bodies. Coulomb had represented magnets as an assembly of small particles conducting magnetic fluids, each containing equal quantities of the two fluids, separated from each other by a medium impermeable to magnetic fluids. Mossotti (' ) imagines in the same way that the di (') Mossotti, /?ec/terc/ies théoriques sur l'induction électrostatique envisagée diaprés les idées de Faraday {Biblioth. universelle, Archives d'électricité, t. V, p. igS; 1847). - Discussione analitica sull' injluenza che l'azione di un 34o BOOK XI. - DIELECTRIC BODIES. lectrics are formed of small conducting bodies, each of which contains equal quantities of the two electric fluids, and which are separated from each other by an insulating medium 5 Faraday (' ) had previously adopted this view. Poisson, by an analysis which is not free of defects, deduced from Coulomb's hypothesis a theory of magnetization by influence. An analysis identical to that of Poisson, applied to the Mossotti hypothesis, leads Clausius (2) to a theory of the electrification of dielectric bodies. Finally, just as Sir W. Thomson made the theory of magnetization by influence safe from any hypothesis on the nature of magnets, we can, following the ideas of Maxwell (^), make the theory of dielectric media safe from any hypothesis on the constitution of these media. In all the studies that make up the first six Books of this Work, the state of electrification of a body is considered to be completely defined when the solid electric density is known at any point inside this body and the superficial electric density at any point on the surface of this body. A similar definition is sufficient, in a great number of cases, to represent the phenomena presented by an electrified body; but it is not always sufficient. In order to include a greater number of electrical phenomena in the representation adopted, we are led to complete this representation by introducing a new class of parameters. From now on, we will admit that, to know completely the state of electrification of a system, we must know : i** At each point of the discontinuity surfaces it contains, the surface electric density; mezzo dielettrico ha sulla distribuzione delV elettricità alla superjizie dei più corpi elettrici disseminati in esso {Mémoires de la Soc.ital. de Modena, t. XXIV, p. 49; i85o). (' ) Faraday, Experimental researches in Electricity, series XI, § 6; Nov. 1887. (") R. Clausius, On the change of internal state which takes place during charging in the insulating layer of a Franklin tile or a Leyden bottle, and on the influence of this change on the phenomena of discharge; i866 {Mechanical Theory of Heat, trans. Folie, i" edition, t. II, p. 86). (') MAxwKLt, Treatise on electricity and magnetism, passim. f CUAP. I. - THERMODYNAMIC POTENTIAL OF DIELECTRICS. 34 1 2° In each point of the volumes of homogeneous or variable constitution in a continuous way which form it: A. The solid electrical density; B. A geometrical quantity, continuously variable from one point to another; we will give this geometrical quantity the name of dielectric induction. We shall seek the form that the internal thermodynamic potential of such a system must have; we shall be helped in this search by the results already obtained in Book IX, Chapter I. The first definitions of the parameters which determine the state of electrification of a system are, in fact, entirely analogous to the first definitions of the parameters which determine an electrified and magnetized system; V dielectric induction simply replaces V intensity of magnetization; but the subsequent hypotheses made on these two kinds of parameters then establish differences between them. These very differences, as we shall see, concern only one point. Let us take up again all that precedes the equality (i i) in § 2 of Chapter I (Book IX), replacing only the word magnetizing intensity by the word dielectric induction and the fictitious magnetic fluid by a fictitious dielectric fluid; then, having arrived at the hypothesis expressed by the equality (i i), let us modify the statement of this hypothesis in the following manner: Let a fictitious distribution be equivalent to Vêlement dv^. It is formed by quantities of fictitious dielectric fluid lia, pi.2, [x'^, ..., [jl!j"' concentrated in pointsM.2^ Mg, M^, ..., M\^ The element dv-i additionally carries free electric charges Pl!j''+", . . . , jx'/' concentrated in points M!f +' ', . . . , Wf. We can write (I) i^.2=x.2+x'i2-+-xi2+---+xVy+2x'iV"-+-^x'A'; y','^' depends only on the mass p.!/"' and on the respective situation of the two points M,, M.j""5 it depends in the same way, WHETHER THE mass [X^"*' IS A FICTITIOUS DIESTRIC FLUID MASS OR AN ELECTRICAL FLUID MASS Following then the procedure indicated in Book IX, Chapter 1, 342 BOOK XI. - DIELECTRIC BODIES. § 2, we will find that we have . /.12 - !^2 ? {' n), then, that we can write (2) \'>^ oAoo ^; dv9 cAo; à(firi^) dv^ àxz ■i.qfù (r) -\~ iq'ta (r') toXn I q, q', ■-', q^P^ being the electric charges distributed over the system, and r, /-', ..., r^P^ the distances of these charges to a point on the element i, ^i, Mi, ^i being the components of the dielectric induction at a point of the element û^p, along three rectangular lines invariably linked to this element (for example its three axes of elasticity); lx^, p,, ... being parameters that define the state of the element dçf . Equalities (i) and (4) of Book IX, Chapter I, and equalities (7) and (8) of this Chapter determine the form of the internal thermodynamic potential of a system containing polarized and electrified dielectrics. This form is the following: (^" = E(r - T2) + WH-'yey (8) This form reminds the form of the internal thermodynamic potential of a magnetized system; it differs only by the presence of the term di> '■f\ d\3 :: - OX relating to the mutual actions of electrified and polarized elements. The internal thermodynamic potential of a magnetized system does not contain any similar term, because it has been assumed that the magnetic elements do not exert any action on the electrified elements. § 2 - Dielectric bodies and pyroelectric bodies. As we have seen (Book IX, Chap. l, § 3), we have, in general, ( H-2Cp23(5V,B,(ïr)î3C + 2C53i(5V,|î,€)C5V+2Cpi2(5V,îî,(i!t)5lU, ), pi, V being three independent quantities of ^, M, C, and the quantities (^pq being functions of ^, M, C, which do not grow CHAP. I. - THERMODYNAMIC POTENTIAL OF DIELECTRICS. 345 not beyond any limit when the quantities %, Î5, C tend to o. But this general form is not, in reality, the most frequent ^ we have seen that, in the great majority of cases, the symmetry of the substance required that one had (lo) X = o, (i"= o, V -G. When, for a substance, these equalities (lo) are verified, we will simply say that the substance is dielectric; when, on the contrary, for a substance, at least one of the three quantities )v, [jl, v is different from o, we will say that the substance is pyroelectric. Among the simply dielectric substances are, in particular, the isotropic substances. For these substances, we have simply (II) g(5i, u, c) = .foiiL), Oro being the dielectric induction and §{dXL) a function of this induction which is such that the ratio ^(31L) does not grow beyond any limit when OÏL tends to o. We shall begin by studying simply dielectric substances with special emphasis on the properties of isotropic bodies; we shall then examine the properties of pyroelectric bodies. § 3 - Transformations of the potential of a system containing dielectrics. Let's take equality (8) which gives the expression of the internal thermodynamic potential of a system containing polarized and electrified dielectrics. We know [Book I, Chap. IX, equality (9)] that we can write integration extending to all volume elements of the system. 346 BOOK XI. - DIELECTRIC BODIES. We have the same (Book VII, Chap. III, equality (17)), Finally, a reasoning analogous to the one which allowed to establish this last formula gives /N r\ 1 ^V I , I r/dY dV d\ dV dY dV\ , (i4) / N^ :j- "'^ = -T- / ■ - I r- -h , ) dv. J II ox \TzJ \âx ôx ây ây dz ()z ) Thus, by virtue of the equalities (12), (i3) and (i4)) (i5) (W^s/I oiva dY dx dv 1 '^^ IL v\d - dv ij I Ox If we adopt the notation defined in Book VII, Chapter III, equality (18), the second member can be written Then, by virtue of equality (10), equality (8) becomes ( .f =E(r- TS)-+-y0^+ - Cu(Y^V)di> --yg(3l, JÔ, C, a, p, ...)rfp. (16) This form of the internal thermodynamic potential will be of frequent use to us. Here is another transformation that will also be useful in many circumstances. Let us imagine, for simplicity, that the system contains a single electrified and polarized body; let us be 1 the space inside this body; 2 the external unlimited space; S the surface of the body. Consider the quantity XI J ^VIL cilg -r- dv. OX I CHAP. I. - THERMODYNAMIC POTENTIAL OF DIELECTRICS. 347 An integral by parts allows to write mIjU ^ rft^ = - Q V[^l, cos(N,,iF) -+- 1)1) cos(Ni,jk)-<- 3 cos(Ni, z)] dS Now we have [Book YIII, Chap. III, equalities (8) and (9)], \ ôx df dz j 411 JL cos(Ni, x) The previous equality becomes (17) /cil.-- flft'::^- - -VV -r^ + - ,- U?S - - - / V AtJ "3?P. J, !l <^37 4t^0 V^Ni (^Nj 47rJi Green's theorem, applied to space 1, gives (18) Ç\ ^-^dv-^-^Y ^d?>^ f'^^ydv^'^^d?). On the other hand, as in all of space 2, we have AV = 0, AtJ = G, Green's theorem gives (19) ô\ Equalities (17), (18), (19) give Ji 11 àx 4^0 \d^i dNiJ 4îtJi Let = 2 1 -T- Ûoils Ji tox dv. di>, oDlL = ^ SJL -I- ^ SDb + ^ 88, d\L DYb OlL CHAP. II. - PROPERTIES OF THE DIESTS. 35l So we have d ,^. ^^ X rf.^(,mo" rft-. This quantity must be equal to o whatever the arbitrary variations SJU, SiJb, §3. If therefore we pose (I) F(Oli) rfif(OlL) 1 0/ifi " -sF(OTl)A(V + -U), Dl, = -£F(01L)^(V + Î)), f o -£F(aiL)^(V + îl). we see that we must have, at any point of the body 1, (2) These are the equalities that define the polarization state of a perfectly soft dielectric. These equalities are analogous to those that define the magnetization state of a perfectly soft body. They can be treated in the same way. Returning to the demonstrations given in Book IX, Chapter VI, we see that, if the function F^DYL) were negative, U equilibrium defined by equations (2) would be unstable. The function F(OIL) is therefore necessarily positive. Therefore, since, according to equation (i), we have (3) ^(DlL)=r F(01L.) dSÏL, we see that the function J^(Oli) is also positive for all dielectrics. By reproducing the demonstrations given in Book IX, Chapter III, we would arrive at the following conclusion: If the quantity F(OIL) decreases as DIL increases, remains independent of DR, or increases weakly with OR, then the polarization state defined by equalities (2) is unique, and corresponds to a stable equilibrium. Let's take a dielectric removed from the action of any electric body 352 BOOK XI. - DIELECTRIC BODIES. The equilibrium equations will obviously be verified if the polarization is assumed to be zero at any point of the dielectric. The equilibrium equations will obviously be verified if we assume zero polarization at any point of the dielectric. However, only one equilibrium state is possible on this dielectric. Therefore, when a perfectly soft dielectric is placed outside an electric field, it has no polarization. Franklin's classic disassembled Leyden bottle experiment shows that glass, polarized in an electric field, can remain polarized when placed out of the field. Matteucci demonstrated the residual polarization of whiting and mica blades. This residual polarization, analogous to the residual magnetization of a piece of iron removed from a magnetic field, can only be explained by attributing a coercive force to the dielectrics studied. § 2 - Specific inductive power. The function F(3TL) plays, in the study of dielectrics, the same role as the magnetizing function in the study of magnets. We will call it the polarizing function. In the study of magnetism, much use is made of the Poisson approximation, which consists of considering the magnetizing function as independent of the magnetizing intensity; similarly, here, it is useful to make the approximation that consists of considering the polarizing function as independent of the polarizing intensity, and to replace it by a constant polarization coefficient k. The equations (2) then become (4) J '\S'o= - tk~{Y-^m, \ oz ^ ' The name of specific inductive power is given to the quantity D defined by the equality (5) D =i-f-4TOA-. CHAP. II. - PROPERTIES OF DIELECTRICS. 363 It is the analogue of the quantity to which Sir W. Thomson gave, in Magnetism, the name to magnetic permeability. § 3 - Electrical equilibrium on a conductive body placed in the presence of an electric current of dielectrics. Let us now imagine a system containing a conducting body, i.e. a body on which the electric charges are likely to vary. This body can, moreover, present a dielectric polarization. Let's find out under which condition the electrical equilibrium will be established on this body. Let us imagine that we modify the distribution on this body without changing its state or its dielectric polarization. According to the equality (28 bis) of the previous chapter, the only variable terms of the internal thermodynamic potential of the system are the terms n^ -(V+2Î1)J<7. . It is easy to see that we have finally a calculation often made gives us We have thus (5) oj" = V [s(v-t-î))-+-e]Sg. This quantity must be equal to o, but not whatever the quantities ùq. Indeed, the variation imposed on the system must leave constant the electric charge of the conductor. We must therefore have 2 oq = o. Therefore, according to a known theorem of the calculus of variations, it D. - II. 23 354 BOOK XI. - DIELECTRIC BODIES. must exist a constant C such that the quantity V[£(V+î")-i-e + C]ô^ is equal to o whatever the quantities ^q are. In other words, /?OMr that V electrical equilibrium is established on a conductive body, it will be necessary to have, in all the points of this body, (6) e(V+ÎI)-He = const. If the body is both a conducting body and a perfectly soft dielectric, equilibrium will only be established on this body if conditions (2) and (6) are both satisfied. Let's consider a system formed : 1° One or more bodies 1, which are both good conductors and perfectly soft dielectrics; 2° Of wine or of several bodies 2, which are perfectly soft dielectric, but perfectly bad conductors; 3" of one or more bodies 3, whose electrisalion state and polarization state are invariable. It will be easily demonstrated, following the methods we have constantly used in the study of electricity and magnetism, that, on a similar system, only one state of equilibrium is possible and that it is a stable state of equilibrium. Let us take a body which is both a perfectly soft dielectric and a good conductor. Conditions (2) and (6) will, as we have said, be both verified for this body. Now, according to condition (6), we have The equalities (2) thus become .%= F(31i)^, d:v \ oz CIIAP. H. - PROPERTIES OF DIELECTRICS. 355 In the case where the body is homogeneous, these equalities reduce to (8) X=o, ^S[> = o, S = o. Thus, in a homogeneous body that is both a good conductor and a perfectly soft dielectric, no dielectric polarization remains at the moment of equilibrium. We have, inside this body, according to equality (6), A(V + 15)=o. But, as there is, inside this body, no dielectric polarization, we have, in any point inside this body, So we also have AV = o. When a body that is both a good conductor and a perfectly soft dielectric is in equilibrium, there is no free electricity in it. We see, therefore, that all the results established in Book II for the electrical distribution on homogeneous bodies which are good conductors remain exact, even when we admit the possibility of dielectric polarization, for bodies which are both good conductors and perfectly soft dielectrics. When we complete the theory of electrical phenomena by introducing dielectric polarization into our reasoning, we do not lose any of the results already acquired. § 4 - Properties of a dielectric insulating blade capacitor. Measurement of the specific inductive power. Let's imagine a capacitor formed as follows: a closed surface S, (^fig. 25) surrounds a good conducting and perfectly soft dielectric body 1, forming the internal armature. Outside the surface Si, a space 2 is empty; it is limited by a closed surface S2, enveloping S,. Between the surface S2 and a closed surface S3 which envelops it is a space 3 filled by a perfectly soft dielectric, but perfectly bad conductor. 356 BOOK XI. - DIELECTRIC BODIES. The surface Sg is followed by a new empty space 4 which is externally limited by a closed surface S^ enveloping the surface Sj. Between the surface S4 and a closed surface S3 which surrounds it is a space 5 filled with the same substance as space 1. Fig. 25. Finally the unlimited space 6, outside the surface S5, is an empty space. The body 4 is put in communication by a wire of the same substance with a very distant body, also of the same substance, maintained at the potential level U (source). The body o is put in communication by a wire of the same substance with a very distant body, also of the same substance, maintained at the potential level o (ground). Body 3 does not contain any electric charge. Moreover, we will admit that it has a dielectric polarization coefficient k independent of the intensity of the dielectric induction. We propose to determine the state of electrification and polarization of a similar capacitor. This determination obviously comes down to that of the two functions V and tH. These two functions are continuous in all space and equal to o at infinity like a potential function. CHAP. II. - PROPERTIES OF DIELECTRICS. 357 The function V admits, in all space, partial derivatives of the first order which are finite and continuous except on the surfaces 84 and S^; it is easy, indeed, to see that the surface S5 is not electrified; it is enough, for that, to replace the polarized dielectric by an equivalent surface layer and to apply the known theorems on the electrostatic induction of hollow conductors. In each of the regions 1, 2, 3, 4, 5, 6, the function V is harmonic. The function M is harmonic in each of the regions 1, 2, 4, 5, 6. It is also harmonic in region 3. In this region, indeed, the equalities (3) are satisfied. If we differentiate the first of these equalities with respect to x, the second with respect to j^, the third with respect to z, and if we add the results obtained member by member, we find If we observe that we have, in this region, AV = o, we see that we have (1 -^ ^Tzzk) AU = o, or else, since "A" cannot be negative, At) = o. Inside conductor 1, according to what we said in the previous paragraph, V will have a constant valueVi and W another constant value tili. Moreover, as there must be a balance between conductor 1 and the source, we will have (9) V, + ll)i=U. In V space % M will continue to have the constant value HIi, because its first derivatives do not experience any discontinuity when crossing the surface S". On the contrary, V will have a variable value. In space 5, V and tll will have constant values. Since the first order partial derivatives of these functions are continuous on the surface S5, these functions will still have the same 358 BOOK XI. - THE ELECTRIC BODY. constant values in space 6. Now, each of them being equal to o at infinity, we see that, in spaces 5 and 6, we have V = o, 1I>=:0. The function tlJ has its first order partial derivatives continuous across the surface S/,. Thus, in V space 4, we have still D = o, while the function V varies. Finally, in V space 3, both functions V and tH are variable. At a point on the surface S2, we have we also have an -4-a3 - ai) 1 - 3(2+ as- ai' (iq) { In region 3 V = U ' I - "2-+- "3- "4 In the region 4 V = U ( W - ai ) 1 - aj -H as - ai ' \ In regions 5 and 6. V = o These formulas (19) determine the electrical distribution on the conductive plate capacitor; they also determine the limiting distribution which is established after a more or less long time on the dielectric plate capacitor. The load of the collector is ^'-~lii!^dW,^'-~J^. I_a2-+-a,-ai0 Jn;"^^' . and, as S (20) Q,== ^ I - "2+ "3- ai In Gaugain's experiments, the surfaces S, S2, S3, S, were parallel planes; from one experiment to the other, the thickness of the dielectric plate was varied, i.e. the distance of the two planes So, S3. But the distance of the planes Si, S2, on the one hand, and the distance of the planes S3, Sj, on the other hand, were, by the interposition of shellac pellets, maintained the same in all the experiments. From then on, in all the experiments, the two binomials (i - aj) and (aa - ol^) had the same values. The limit charge Q, should, under these conditions, be independent of the thickness of the dielectric and its nature; it should be identical to the charge immediately obtained by replacing the bad conductor by a good conductor. The results obtained by Gaugain are in accordance with these indications of the theory; here are these results: the load of the collector is 366 BOOK XI. - DIELECTRIC BODIES. measured by the number of sparks obtained by discharging it through the electrometer-gauge. Interposed disc. Time. Zinc After a few moments After a few moments After a 6 minute charge After another charge of lo™... After a few moments of charging After another 5" charge " " II Load from collector. Stearic acid (6'™ thick) Stearic acid (17"" thick) Gutta-percha (thickness 6"") Shellac (6""" thick) " " 20 .. " " 2'' iS".. " " î'Mo".. After a charge of a few moments After a new charge of 5*"... " " 10™. " " 20'".. After a charge of a few moments After a further charge of 5™... " " 20™. . Sulfur (6"" thick). After a charge of a few moments After a charge of 7''3o™ 21 i3 20 21 9 14 14 16 >9 ■?.o i5 17 9.1 22 14 20 22 i5 21 2° The difficulty that we have just mentioned is not the only one that the determination of specific inductive powers presents. If, as in Cavendish's and Faraday's experiments, the dielectric is in contact with the armatures of the capacitor, and if, moreover, the dielectric is not perfectly insulating, there will be penetration of electricity into the dielectric, contrary to the conditions assumed by the preceding theory. 3 " Finally, the preceding theory assumes that the dielectric is perfectly soft, which, as we have seen, is far from being exact for all. § 6 - Permanent current in a dielectric. If a dielectric is not a perfect bad conductor, it can be the seat of electric currents. What will be the laws CHAP. II. - PnOPRlETIES OF THE DIELECTRICS. 367 What will similar currents do when they reach steady state? We will base the establishment of these laws on the hjpothesis which has constantly served us, in JJvres V and VI, in the study of this kind of questions. We will calculate the expression of the internal thermodynamic potential rf that the system would have if, stopping at a given moment all the currents, we kept the system in its electrified and polarized state. To a charge dq placed at a point M, we will give any infinitely small translation (5.r, 5/, 03). The quantity i will undergo a variation ùi. We will admit that we have, whatever Sx, 8^, Ss, u^ t', w being the components of the electric flux at point M and U the specific electric resistance at this point. Let's apply this hypothesis. According to the expression (28 bis^ of the internal thermodynamic potential, we have S^-j ^[KV + tJ)+e]8;r ^ [£(V + lH)+e]07 ^^U(y^^)^^\lz[dq. to d Our hypothesis will therefore lead to the following equations '>êiU = dx ' dx ày ày ae ()(V-f-î)) U (V == - z ~ - - - dz dz Moreover, if the dielectric is perfectly soft, the equalities (2) are satisfied at all points. Let us limit ourselves to studying a homogeneous dielectric, perfectly soft, having a constant polarization coefficient. In these conditions 368 BOOK XI. - DIELECTRIC BODIES. ditions, the equalities (21), joined to the equality of dv dw T" "-" T~ "*" IT" = O' ox ay oz which expresses that the current is uniform, give {%i) A(V + î))=o. On the other hand, the equalities (4) give or because of equality, dX d\JÎ) dB I .^ - 1 ■ -1- - - = - AtH, dx dy dz ^iz (23) (n-4TOA-)AD4-47t£/cAV = o. k can never be negative; therefore, the comparison of (22) and (28) gives the two equalities (24) ' AV==o, (25); AV = 0. Equality (24) teaches us that, in a homogeneous, perfectly soft dielectric with a constant polarization coefficient, through which a permanent current flows, the polarization is both solenoidal and simple lamellar. Equality (26) teaches us that in V inside this dielectric there is no free electricity. The comparison of the equalities (6) and (21) gives, for a homogeneous dielectric X = kViu, ( G =k'^w. According to these conditions, in a similar dielectric, the polarization density is, at each point, directed like the electric flux; it is proportional to it in magnitude; the proportionality coefficient is the product of the dielectric polarization coefficient and the specific resistance. The equations we have just given essentially assume that the steady state is established; they cannot be used, I CHAP. H. - PROPERTIES OF ELECTRICITY. 36o at least without prior justification, to study the variable state of a dielectric plate capacitor. We could extend indefinitely the study of dielectric bodies and take up for them most of the questions we have dealt with for magnets; we will not dwell on this; we will leave it to the reader to extend to dielectric crystals what we have said about magnetic crystals ( *). (' ) See on the subject of the dielectric properties of crystals: Jacques Curie, Recherches sur le pouvoir inducteur spécifique et la conductibilité des corps cristallisés (Annales de Chimie et de Physique, 6° série, t. XVII, p. 385; 1889). D. - 11. 34 370 BOOK XI. - DIELECTRIC BODIES. CHAPTER m. ATTRACTIONS OF ELECTRIFIED BODIES IMMERSED IN A DIELECTRIC MEDIUM. § 1 - Electrical distribution on conducting bodies immersed in a dielectric medium. While leaving it to the reader to extend to dielectric bodies the theories that we have developed when studying magnetic bodies, there is one question that we will examine here in detail, because it gives rise to remarks that the study of magnets did not lend itself to. We have seen (Book IX, Chap. VIII and IX) that the properties of diamagnetic bodies can only be explained by admitting, in the spaces that seem to us to be empty, the existence of a fluid endowed with magnetic properties; the experiments of M. P. Joubin (Book IX, Chap. VIII) even seem to indicate that this fluid is endowed with coercive force. If we admit the existence of a similar fluid endowed with magnetic properties, it is natural to suppose that it could also be endowed with dielectric properties; all the electrified bodies that we observe would thus be immersed in a polarizable medium. Then arises the following question whose importance, great by itself, increases still more by the role it plays in the discussion of certain theories: If we suppose that the mutual actions of electrified bodies immersed in an ideal insulating medium, incapable of electrifying or polarizing, are given by Coulomb's laws, CHAP. III. - action of a dielectric medium. Syi What disturbance does the presence of a dielectric medium in which these bodies are supposed to be immersed bring to the consequences of these laws? It is this question that we will examine in this Chapter. Let's start by studying the laws of electrical distribution on conducting bodies immersed in a dielectric medium. We will denote by V the potential function of the electrical distribution, and by tU the potential function of the dielectric distribution. The equilibrium condition on a conducting body will then be (i) £(V-f-tJ)-t-0 =: const. The electric density at a point of the separation surface of the conductive body and the dielectric medium has the value ~ ^Tt \dni diie As equality (i) gives drii dni z dn,- ' the previous relation can be written ^T^ \àne drii ) ^r^z d/ii On the other hand, by designating by X, iJÎ), G the components of the polarization intensity, we have, at any point of the considered separation surface -7 1- ^ - = 4i^[- c)(Y + |i). (6) a = ^-^~~. Consider the function (i + 47r£A-)(V + 11). This function is constant inside the conducting bodies; according to the equalities (6), it verifies Laplace's equation outside the conducting bodies; it thus represents the potential function of a distribution which would be in equilibrium if the conducting bodies were immersed in a non-electrifiable and non-polarizable medium; at a point on the surface of these conductors, the density would be for value J + 47r£A- ^V -+-t>) 4 71 in Ile i.e. o-, according to equality (6); hence the following conclusion: With the restrictions indicated, V electricity is distributed over a system of conductive bodies immersed in a dielectric medium as if this medium were replaced by a non-polarizable medium. CHAP. III. - action of a dielectric medium. SjS This theorem extends to the case where the medium contains a perfectly soft dielectric body, devoid of any electric charge, and whose polarization coefficient is constant. In this case, in fact, the functions V and liD verify the equalities given in the above; moreover, inside the dielectric body considered, we have AV = o, AV = o, and, on the surface of this body, dY d\ orii aue (1-4- 4tc£A-)-- -t- (1 + \izzk )-T \- l^Tztk h4toA- - - = 0. diie orii one orii By virtue of the penultimate equality, the latter can be written The function U = (i 4- ir^ek) (V -t- tiî) thus satisfies the following conditions: It is continuous in all space; It is harmonic inside the dielectric body and in the space outside the dielectric body and the conductive body; It is constant inside the conducting body; It verifies, at the separation surface of the dielectric and the medium, the equation i + 4Tr£A-' dU toV _ iH- 4 iî£ A: drii drig Consider a quantity y defined by the equality . , , l-i-^TZtk' (7) l-l-4Tr£Y= -, 7' Inside the dielectric body, imagine a polarization whose components have the value (8) \ B=-EY^=-sY(i+4^sA-)|;(V-+-îl), G = -eY^ =-£Y(i + 4^^A-)^(V4-î)) 374 BOOK XI. - DIELECTRIC BODIES. at the surface of the conducting body, let us place a distribution whose density has the value 47: drii 4''^ ^^t (V + tH). This distribution, it is easy to see, would have as its potential function, both electric and dielectric, the function U; moreover, it would ensure the equilibrium of the system supposedly immersed in an ideal, non-electrifiable and non-polarizable medium. Now the density of the electric layer that the conductive body carries in this distribution is identical to the density of the layer it carries when it is immersed in the polarizable medium. When the dielectric is immersed in the polarizable medium, it takes on a polarization at each point whose components have the value (9) If we compare these equalities (9) with equalities (8), we find A = ^, (j-}-47r£X-)ol,, B = ■l(n-4Tr£Â-)lll>, C = 1,(1 + 4 TTC A-) S, l o/l) = -s^'^(V + î.), ..= 1 --zk'~{y^v). or, according to the relation [n), (10) ol, = k' A /.' - k'^' Dî> = k k' 7c '^' __ k' -, C. k'-k This leads to the following proposition: Suppose we want to find the distribution of CHAP. III. - action of a dielectric medium. 3y5 We can eliminate this dielectric medium, search for the state of equilibrium of the conductors and dielectric bodies, by assigning to each of them not its true polarization coefficient, but a fictitious one defined by the polarization coefficient of the conductors and the dielectric bodies. We can remove this dielectric medium, and look for the state of equilibrium of the conductors and dielectric bodies, by assigning to each of them not its true polarization coefficient, but a fictitious coefficient defined by the equality {'j). We will then obtain, on the conductors, the electric density we are looking for and, on the dielectrics, a polarization whose components can be multiplied by p - y to obtain the components of the polarization we are looking for. When the system contains not only good conducting bodies or perfectly soft dielectric bodies, but also bad conducting electrified bodies or dielectric bodies endowed with coercive force, there is no longer any simple rule allowing the study to be reduced to that of a system immersed in an ideal insulating medium. § 2 - Attractions between bodies immersed in a dielectric medium. Let's consider various electrified bodies, all of which occupy a space designated by the index 1, and various dielectric bodies, all of which occupy a space designated by the index 2, immersed in a dielectric medium which occupies the unlimited space 3 around them. According to equality (i6) of Chapter I, if we assume that all the dielectrics considered are isotropic, the internal thermodynamic potential of the system will have the expression We will transform this equality by assuming the following: 1° Dielectric bodies are perfectly soft and not electrified. 376 BOOK XI. - DIELECTRIC BODIES. 2° Their dielectric induction coefficient does not depend on the polarization intensity. Let /r be the dielectric induction coefficient of the medium; let k' be the dielectric induction coefficient of the dielectrics immersed in this medium. We will have '2 A. % K Besides, the equalities ertas - - oA r J oiVuo - SA. r > ax OX *^3 - S/l ■)■ O2 - £/t r az oz give Oîl|= £2A'2 n(V + îl), ^L|= £2/t'2n(V-i-îl). If we observe that k and k' are positive, these equalities give '\Yt 2 ^2 '^TT 2 ^2 ^5 = -A'n(V-Hiii), :^ = '_/c'n(V + ti)), and we have (II) + g^(i+4TOA-) Y being defined by the equality (7). To the restrictions already made for the proof of this equality (i i), let us now add the following: i" We can neglect the variations that the quantity undergoes from one point to another of the same electrified body. 2° Electrified bodies are good conductors. Let A, B, ... L the various electrified bodies of the system; let 0,^, 0^, . . . 0l the values of the quantity which correspond to them, values which are invariant if the state of these conductors does not change; let Q^, Qc, ..., Ql the invariant charges of these conductors. r CHAP. III. - action of a dielectric medium. 877 ductors. We will have 2^5^ = Qa Qa -H e" Qb +. . .-t- and Ql . The second member is obviously invariant. The quantity / Qy being a constant, we can make it disappear from the expression of the internal thermodynamic potential. If all the electrified bodies which compose the system are good conductors, we will have, for each of these bodies £(V -t-î))-t- e = const, or, since we neglect the variations of 0,. V + U = const, which requires that we have, at every point of the region , n(v + U) = o and, therefore, (12) fu{Y-\-V)dvi^o. Let's put (i3) U = (h-4toA")(V4-Î1). We will have f =(' + 4TSi)|;(V+1.), Let's square the two members of these equations and add member by member the results obtained; we will have nu = (i+4 7r£A:)2n(V + î>), and the equality (i i) will become, taking into account the various remarks we have just made, Let's give the system an infinitely small displacement; the forces 378 BOOK XI. - DIELECTRIC BODIES. the internal thermodynamic potential will undergo a variation S.f; as the electric and dielectric equilibrium is established on the system, we will have dïBi -)- 83" - o or ^5,- = - 3E(r - TS) 4 ioT fnUdvs-^ ( {1+ i'Kty)UU dvA 41TSA- LJ3 J^ J The quantity oE(r - TS) represents the work of the internal forces that would act in the system in the neutral state. The quantity (,4) d(^', = -- ' TT 1 -h l\TZS.k fuUdv3-}- r(i^-^Tzzy)UUdvA represents the work of the electrical forces. Consider a system constituted as follows: It is formed by conductors and dielectrics having the same shape and position as in the previous system. The environment in which they are immersed is impolarizable. The dielectric induction coefficient of dielectrics is the quantity v, defined by equality (7). On the conductive bodies, we distribute electricity in the same way as in the previous system. On dielectric bodies, the components of the polarization are related to the components A, itl, G of the polarization in the previous system by the relations (10) According to what we have seen in the previous paragraph, such a system is in equilibrium; the potential function of both the electrical and dielectric distribution it carries is U. Reasoning as we did in the previous case, A = k - k lIj k' a\a, B = k ~P k iPo, G = k k k CH.VP. m. - ACTION of a DIELECTRIC ENVIRONMENT. 379 we would find that, when a certain variation is imposed on this fictitious system, the internal forces of electrical origin elTect a work . (i5) d(B'i = -^-^\ y^nU^/t's-^ /"(H- 4 ti^y) nuirai Comparing equalities (i4) and (i5) gives this relationship d^'i= (i-{-4Tzzk)d(E "i, which leads to the following proposition: The electrical actions cjiii acting in the considered system are obtained by multiplying by (^\-\~^Tzzk) the magnitude of the forces acting in the fictitious system cjuc we just studied, without changing the direction of these forces. This theorem allows us to compare the actions of a system immersed in an impolarizable medium with the actions of the same system immersed in a polarizable medium whose polarization coefficient is A'. The presence of the polarizable medium does not change the direction of the actions that occur between two given conductors carrying given charges; it multiplies the magnitude of these actions by i-\-^Tzzk It does not change the direction of the actions that are exerted between a given conductor, carrying a given charge, and a dielectric body whose dielectric induction coefficient A' is given; it multiplies the magnitude of these actions by I k' I -f- 4t-£^ k' - k It does not change the direction of the actions that are exerted between two dielectric bodies with given induction coefficients A' and A" and that are polarized by the action of given conductors, carrying given charges; it multiplies the magnitude of these actions by I k'k" ^Tzzk {k'-k)(k"-k) ■^80 BOOK XI. - DIELECTRIC BODIES. We can still give a very simple and interesting interpretation of the previous theorem. Let's say (17) i + 4t:e'y'= i-+-47r£Y. By virtue of equalities (7) and (16), the latter equality becomes (.8) y'=>^'-/ By virtue of equalities (16) and (17), equality (i4) becomes Moreover, by virtue of equalities (7) and (i3), equalities (9) become li'd = - z'k-r-, oz These equalities, together with equalities (10) and (18), give (9.0) ^B = -sY-, U is, as we have seen, the potential function of an electric layer, of density a-, distributed on the surface of the conductors, and of a polarization, of components A, B, C, distributed on the dielectrics. The equalities (20) lead to the following conclusion: If the dielectric medium were eliminated, if the constant s of Coulomb's laws were replaced by the constant e', if the induction coefficient k' of the dielectric bodies were replaced by the coefficient y', the distribution of electricity in equilibrium on the conductive bodies would not be changed, and the polarization in a CHAP. III. - ACTION OF A DIELECTRIC MEDIUM. 38 1 point of a dielectric body would have for components the quantities A, B, G. This first result obtained, the comparison of equalities (i5) and (19) leads to this fundamental theorem: In a dielectric medium whose induction coefficient has the value /r, are immersed conductors carrying determined charges and dielectric bodies having determined polarization coefficients. The electric and dielectric distribution on this system and the actions that are exerted in this system are those that Von would calculate if, disregarding the existence of the dielectric medium, the constant e of the electrostatic actions was not given its real value, but the fictitious value e'^ , , and, for each dielectric body, not its real induction coefficient, but a fictitious coefficient equal to the excess of its real coefficient over the coefficient of the medium. This theorem assumes that all electrified bodies in the system are good conductors and that the only dielectric bodies are perfectly soft bodies. This last restriction is the reason why an analogous theorem cannot be found in the study of magnetism, since the systems studied cannot be composed exclusively of perfectly soft bodies. This theorem shows us that, if we admit that the vacuum is susceptible to polarization, we are not obliged to change anything in the laws of electrical distribution or actions; we only have to modify the value of the fundamental constant of Coulomb's laws and of the induction coefficients of dielectric bodies. It follows that it is not possible to determine the value of the constant £, nor the value of the inductive power k of the vacuum, but only the value of the quantity Does the constant k have a significant value? Does the measurable quantity e' differ significantly from the value that should be 382 BOOK XI. - DIELECTRIC BODIES. to attribute to the quantity inaccessible to experience e? It is not possible to answer this question in a formal way. 11 is, however, a remark that makes the following opinion likely: the induction coefficient of the vacuum is a quantity that is either zero or very small; the constant t' differs little from t. We have seen that the apparent induction coefficient of a dielectric was related to its true induction coefficient by the relationship Y'= ^'_ k. If k had a significant value, there is no reason why y' could not have a significant negative value for certain bodies; such bodies would be to dielectric bodies what diamagnetic bodies are to magnetic bodies; they would be dielectric bodies. Thus, if the vacuum had a significant induction coefficient, we would probably observe dielectric bodies whose apparent cV induction coefficient would have a significant negative value. However, experience has never revealed the existence of any dia-dihedral bodies; if such bodies exist, they must have a very low apparent polarization coefficient. This fact agrees well with the hypothesis that the vacuum has a very low polarization coefficient, without, however, demonstrating the accuracy of this hypothesis. Similarly, the absence in nature of any strongly diamagnetic body suggests that the magnetization coefficient of a vacuum is a very small quantity; this is what we have consistently admitted in previous books. This hypothesis that the vacuum is only weakly dielectric and magnetic is all the more valuable for the theory that, in the experiments of M. Paul Joubin, the vacuum seems to be endowed with magnetic coercive force; it is therefore possible that it is also endowed with dielectric coercive force; if its magnetic and dielectric properties were notable, the whole theory of electricity and magnetism would be called into question. PYROELECTRIC CRYSTALS. 383 CHAPTER IV. PYROELECTRIC CRYSTALS. § 1 - The equilibrium state of a homogeneous hemimorphic crystal. In the previous chapters, we have studied the equilibrium state of bodies whose structure verifies the conditions expressed by the equalities X = O, |JI. =^ O, V = o. We will now free ourselves from this restriction and study hemimorphic dielectric bodies. We will only assume, in order not to complicate our calculations, that the studied crystal is homogeneous. Let's take for coordinate axes Ox, Oj, Oz the directions of the elasticity axes of our crystal. If this crystal is a perfectly soft dielectric, reasoning as we did in Book X, Chapter I, § 6, we will find that we must have, at any point of this crystal In this equality, the quantities T and Opq are functions of -X, i)b, 3, related to the coefficients of the dielectric induction surface, as discussed in Book X, Chapter I, § 4. These coefficients satisfy the condition 384 BOOK XI. - MORE DIELECTRIC BODIES. If the studied crystal is, moreover, a good conductor, one must have, at any point of this crystal, according to the demonstration given in § 2 of the preceding Chapter, £(V4-11l) + e = const. or, because of the homogeneity of the crystal, (2) £(V-t-tII) = const. The condition (2) simplifies extremely the equalities (i), which become X =- Tji (SiiX -}- 012p. + Ô13V), (3) . I !">=- ;jî ( 821 X + 8221^+ Ô23V), G =- Tp (831X + 832(^-1- 833V). The quantities \, pi, v have the same value at any point of the homogeneous crystal; the quantities T and Zpq are, at any point of this crystal, the same functions of X., al\), S. These equations (3) will thus lead to take for Jl, ilb, G the same values at all points of the crystal. Hence the following conclusion: When V equilibrium is established on a hemimorphic and conductive crystal, it presents, in all its mass, a uniform dielectric polarization which depends only on its nature and its temperature. If we have, in the whole mass of the crystal, aHo = const, oPd = const, G = const, we also have dx dy dz and, consequently, Equation (2) then gives AV= o; hence the following conclusion: When V electrical equilibrium is established on a bound crystal CHAP. IV. - PYRO-ELECTRIC CRYSTALS. 385 viimorphic and conductive, it does not contain free electricity inside. Let's find out how electricity is distributed on the surface of this crystal. The surface density of electricity is given by the formula But equality (2) gives d^i " ~ os] ' We can therefore write noting that TZ. ( ^."^^j = Xcos(N/, a^) + -1)1.008 (N,-, 7) + 3 cos(N/, z), we see that we have ( 4 ) a = a' -f- or" with / a' = -A., cos(N,-, J7) -f- ill>cos(N-,7) -H 3 cos(N/, z), It is easy to give an interpretation of these two densities a-', a/' defined by the equalities (5). The polarization of the crystal being uniform and, consequently, solenoidal, we know (Book VII, Ghap. IV, § 2) that the action that this polarization exerts on the points outside the crystal is equivalent to the action of a surface layer having for density - [X cos(N/, a7)-i-i)î)Cos(N^, 7) + Gcos(N/, z)]. The density cr' is equal and of opposite sign to the previous one, so that the action, at the points outside the crystal, of the layer which has this density, destroys exactly the action exerted at the same points by the polarization of the crystal. To interpret the density a", we will notice that the function ("Il -f- V), continuous in all space, admits inside the D. - II. 25 386 BOOK XT. - THE BODY OF THE TRIQURS. crystal a value conslanle A and is harmonic outside the crystal. Therefore, we see that the density a" is that of a layer which, in equilibrium of itself on a conductor of the same shape as the crystal, would bring this conductor to the potential level A. Let's summarize these findings: A heinimorphic crystal which is a clear conductor in equilibrium has an electric layer on its surface. This layer can be seen as formed by the superposition of two other layers. The density of the first is, at any point, equal and of opposite sign to the density of the fictitious layer which is equivalent to the polarization of the crystal. The second is the layer cjui, in equilibrium of itself on a conductor of the same shape as the crystal, would communicate to this conductor a potential level equal to the constant value of (V+ tîl) inside the crystal. The action of the crystal at the external points is reduced to the faction of this last layer. Let us suppose, in particular, the crystal in communication with the ground; we have then V 4- 11=0, and the last layer disappears. If we observe that all crystals are more or less conductive, we can easily reach the following conclusion: If a hemimorphic crystal, whose temperature is kept constant, is put in communication with the ground for a sufficient length of time, it will no longer show any signs of electricity. Of the two layers into which the crystal can be decomposed, the first has zero mass; for the second to have zero mass, it must have zero density at all points. If, therefore, no electric charge has been communicated to a crystal, if it is kept at a constant temperature and isolated, it will always cease, after a sufficiently long period of time, to manifest any electric action. CH\P. IV. - PYRO-ELIC CRIST.VUX. 387 All of the above is general; let us now adopt, for the dielectric phenomena presented by crystals, an approximate hypothesis similar to that which we have adopted for magnetic phenomena and which we have named Poisson's approximation (Book X, Chap. II). Let us take as coordinate axes the principal axes of dielectric induction; let vji, Wo, ^s be the principal coefficients of dielectric induction; let ^, JE5, C be the components of the induction related to these axes; we will have, instead of the equalities (3), the equalities (6) < î" = - 7!l2[Jl, It is therefore easy to determine the direction of polarization of the crystal. Let us take X inverse ellipsoid of dielectric induction of the crystal, surface which has the equation TJTi;--+- CT2rj2-f- TJTa^^::^ I. Through the center H of this surface, let us lead a vector rajon whose director cosines are proportional to - )^, - [jl, - v. \The dielectric induction of the crystal is normal to the conjugate plane of this direction. If the crystal is a uniaxial crystal, and if we take for X axis the axis of the crystal, we will obviously have, by reason of symmetry, and the equalities (6) will become ( 5l = - T^iX, (7) jlî=o, 1 C = o. The polarization will then be directed along the axis of the crystal. § 2 - Pyroelectric phenomena. For a given crystal, the quantities X^ \)î), 3, given by the equations (3), are functions of the temperature. Let us denote 388 BOOK XI. - DILITIC Bodies. by T the absolute temperature, and by .1, (T), ill) (T), S (T) the values of these functions at temperature T. Let's imagine that we take a crystal that has been in communication with the ground for a long time and no longer shows any trace of electricity. Keeping it carefully insulated, let us bring it to a certain temperature T and maintain it at this temperature for a very long time, so that equilibrium is established. The crystal will then carry, on its surface, an electric distribution whose surface density a- (T) will have the value, according to the equalities (5), (To), ^Î3(To). G(To). But, on the contrary, each element of the surface of the crystal will keep appreciably, during a certain time, the electric charge that it carried at the temperature T. As a result, the electric surface layer will no longer exactly cancel the action of the polarization of the crystal at the points outside the crystal, and the crystal will show electric actions which, in the long run, will disappear. Let us neglect the slight change of shape that the surface of the crystal undergoes when this one passes from the temperature T to the temperature Tq. At the temperature To, one can look at the surface layer as still having for density (at least in the first instants) a(T) = X{T) cos(N/, x) -i- iJbCT) cos(N,-, j-) + a(T) cos(N,-, z), while the polarization of the crystal will exert the same external actions as an electric layer with surface density - [X(To) cos(N/, x) -h 1)1 (To) cos(N,-,^) -+- e(To)cos(N/, z)]. CIIAP. IV. - PYRO-ELECTRIC CRYSTALS. iSg The crystal will therefore exert the same external electrical actions as an electrical layer, distributed on its surface and having the following density / i:(Ti,To)= [olo(T)--V..(To)]cos(N,-,^) (8) ^-[itï,(T)-"lï>(To)Jcos(N,-,.x) ( +[8(T)-3(To)]cos(N,, z). We have seen (Book VIT, Chap. IV, § 2) how one can obtain, in a very simple way, a geometric image of the density S(T, T"). Let us consider the surface S of the crystal; let us give it an infinitely small translation parallel to the geometrical size which has for components -^(T)-.Â,(To), 1II.(T)-. i)J,(To), e(T)-3(To); let S' be the new position of the surface S. Between S' and S is an infinitely thin layer of which one part is inside the surface S, while the other part is outside it. Let us fill these two parts with electric fluid whose density has everywhere the same absolute value, but is positive in the first part and negative in the second. The density 2(T, Tq), defined by equality (8), will have at each point of the surface S the same sign as this layer and its size will be proportional to the thickness of this layer. In particular, let us assume that the crystal is a uniaxial crystal; let us take the direction of the crystal axis as the x-axis. We will have ill,(T) = o, a)î,(T") = o, 3(T)--=o, €(To)-o. Let us suppose, to fix ideas, that we have .l,^T)-A^(To)>o. Let us cut in this crystal a right prism, whose edges BB are parallel to Oo:. From the above, at temperature Tq, the base B will appear to carry negative fluid with density -[cl,(T)-.l.(To)]. Sgo book xi. - dielectric bodies. The apparent load of this base will be, by designating by S its surface, _[A,(T)-oA.,(T")]S. The base B' will apparently carry positive fluid in equal amounts. The side faces of the prism will not appear electrified. Experience has long since overtaken these indications of theory. Towards the end of the 18th century, the Dutch brought back from Gejlan a curious stone that the natives called tournamal, i.e. ash-puller, because of its property of attracting ashes when thrown into the fire. It is the stone that today, in mineralogy, bears the name of tourmaline. Lemerj gave the first description in 1717. In 1756, OEpinus showed that the properties of tourmaline are due to a development of electricity, and that the two ends of a heated crystal are always opposite, one positive and the other negative. Bergmann later showed that the opposing electricities at the two ends of a heated tourmaline are always equal in quantity. In 1709, Canton (^) observed that the electrification of a tourmaline is not due to the temperature itself, but to its variations. A crystal of tourmaline, when brought to a certain temperature and returned to a neutral state, will remain in a neutral state as long as the temperature does not change; but it will electrify immediately if the temperature changes, and the electrification effects obtained will be in the opposite direction depending on whether the crystal is heated or cooled. This observation is of the utmost importance; it is perhaps the most important one that has been made in this order of phenomena, the particular character of which it marks. The pjro-electric phenomena are not peculiar to tourmaline; they have been observed, as we shall see, on a large number of crystals, but to a lesser degree. (' ) Philosophical Transactions, Vol. LI, p. 4o3. ClIAP, IV. - PVRO-ELECÏRIQIJES CRYSTALS. Î9I The theory of pyroelectric phenomena was first sketched by Sir W. Thomson (*) and more completely treated by Mr. Ed. Riecke (-). We have given, for our part, a theory of pjro-electric phenomena (-'). This theory, as we have done Mr. Lorberg notes, is completely wrong. / § 3 - Gaugain's experiments. We have seen in the previous paragraph that the apparent charge carried by one of the bases of a tourmaline pris'me cooled from the temperature T to the temperature To has the absolute value Q-[.l.,(T)-.l.(T")JS. . This charge is proportional to the section of the prism and independent of its length; so that the quantity of electricity that a battery of tourmalines associated by their poles of the same name can develop is equal to the sum of the quantities that would be provided by the separate elements, and that a battery of superimposed tourmalines develops exactly the same quantity of electricity as any one of the elements which serve to form it. The amount of electricity that a tourmaline produces, when its temperature rises by a given number of degrees, is precisely the same as that which would result from an equal lowering of temperature. But the sign of the two poles of the tourmaline is reversed. These various proposals were obtained by Gaugain (^) in a very careful experimental study of pjro-electric phenomena. (') Sir W. Thomson, On thermoelastic , thermomagnetic and pyroelectric properties of matter {PhilosophicaL Magazine, 5' series, t. V, p. ^4; 1878). (") Ed. Riecke, Ueber die Pyroelectricitât des Tur malins {Wiedemann's Annalen der Physik und Chemie, t. XXVIII, p. 4^; 1886). (') Annales de l'École Normale supérieure, 3' série, t. III, p. 263; 1886. (*) Gaugain, Mémoire sur l'électricité des tourmalines {Annales de Chimie et de Physique, 3° série, t. LVII, p. 5; 1859). Sga, BOOK XI. - DIELECTRIC BODIES. § 4 - Naturally and accidentally pyroelectric crystals. The pjro-electric phenomena are not peculiar to tourmaline; they have been observed in a large number of crystals. Canton had already recognized the same properties in topaz and Brazilian emerald; Brard in axinite; Haûy in boracite, mesotype, zinc oxjde, prehnite, sphene; Brewster in many other natural minerals such as quartz and in several artificial crystals, among which are double tartrate of potash and soda and tartaric acid. All these researches led to an important correlation between the crystalline form and the pyroelectric phenomena. Haûy (') was the first to notice that the crystallized substances which possess pjro-electric properties, such as tourmaline, calamine and boracite, deviate from the law of symmetry and are struck by hemihedron. This observation was then confirmed and generalized in particular by the numerous experiments of Riess and Rose (-). In 1866, M. Hankel (*) published a long series of researches on the pyroelectricity of crystals. These investigations led to the paradoxical consequence that the axis of pyroelectricity is a straight line determined in direction and position, whereas in crystallography all straight lines are determined only in direction. Mr. Friedel and J. Curie ('') showed that the singular phenomena observed by Mr. Hankel were to be explained by an irregular heating of the studied crystals, which, by altering the symmetry of the structure of the crystals, endowed them with an accidental pyroelectricity. MM. Friedel and J. Curie pushed this research further; (' ) Hauy, Traité de Minéralogie, t. III, p. 54. - Observations on the electrical properties of magnesio-calcareous borate (Annales de Chimie et de Physique, i" série, t. IX, p. 69; 1791). (') Riess and Rose, On the Pyroelectricity of Minerals [Archives of Electricity, vol. III, p. 585). (") See G. WiEDEMANN, Die Lehre von der Elektricitàt, 11^ Bd., p. 820. (*) Ch. Friedel and J. Curie, Sur la pyro-électricité du quartz (Bulletin de la Société miner alogique de France, t. V, p. 282; 1882). CHAP. IV. - PYUO-ELLICTBIQUE CRISTAUX. SqB they showed that accidental pyro-electricity was extremely frequent, that it was the real cause of a lot of phenomena attributed to normal pyro-electricity. Starting from this principle, to which MM. P. and J. Curie had been led by experiments which we will discuss in the next chapter, that a crystalline plate develops a quantity of electricity proportional to the dilation of the substance in the direction normal to the plate and to the cosine of the angle that this direction makes with the axis of hemihedral, they did not hesitate to declare that the pyroelectricity of quartz, of cubic substances, such as blende and soda chlorate, which present the tetrahedral hemihedron or the tetartohedron, was purely accidental and due to an irregular embossment. Indeed, the experiment showed that these substances gave no sign of electricity when heated regularly (' ). The study of the pyroelectricity of boracil gave further confirmation to the ideas of Friedel and Curie. The ideas of Messrs Friedel and J. Curie were exposed by M. Mallard in his Traité de Cristallographie (-). M. Mallard deduced the necessary condition for a crystal to enjoy normal pyroelectricity: The crystal must have no center and must present only one axis of symmetry. This rule is consistent with the previous theory. Indeed, for a crystal to be pyroelectric, we must not have at the same time X == O, [i. = O, V = o. If one remembers what was said above (Book X, Chap. IV, § 1), one is immediately led to the rule of M. Mallard. We have seen that a substance whose structure admitted a center in the natural state still had a center after a slight deformation; so that, for this slightly deformed substance, we still have X = o, [JL = o, V = o. (') Ch. Friedel and J. Curie, 5m/- la pyro-électricité du quartz {^Bulletin de la Société miner alogique de France, t. V, p. 282; 1882). - 5m/- the pyroelectricity in blende, sodium chlorate, and boracite (Jbid., t. VI, p. 191; i883). - On the pyroelectricity of topaz (Ibid., vol. VIII, p. 116; i885). (") E. Mallaud, Traité de Cristallographie, Vol. II, p. 171. 394 BOOK XI. - DIELECTRIC BODIES. If we admit that an irregular heating produces only small deformations, we see that an irregular heating will not be able to make this substance pjro-elliptic. We thus arrive at this new rule: The accidental pyroelectricity can present itself only in the hemihedral substances deprived of center. CIIAP. V. - THE PIKZO-lÎLECTRlIC CRYSTALS. SgS CHAPTER Y. PIEZOELECTRIC CRYSTALS. § 1 - Release of electricity by the compression of substances crystallized. Pierre and Jacques Curie (' ) showed that electricity could be released from hemihedral crystals not only by variations in temperature, but also by variations in pressure. Their research became the starting point of the study, today very extensive, of the piezoelectric phenomena. We will indicate here some theoretical principles which allow to clarify this study. Let us consider a crystal in its natural state, and let us take for coordinate axes any three rectangular lines O^, Oyi, OX^. The magnitude (à, jji, v) has, with respect to these three lines, components Xq, p^o> '^'o- If the crystal is homogeneous, we know that it has a uniform polarization inside with components Xo = - l|;(OiiXo-+- 0,2^0+ ÔisVo), (0 { l''"0 = - ^(021X0-^ O22[-l0-^ 023 Vo), xO = Y^°=^''""^ 032[-".a-H Ô33V0). The crystal, put in communication with the ground, when the electrical equilibrium is established, presents at its surface a density (2) ao= X(, cos(Ni-, x) H- 1)1)0 cos(N/,7) -^ SoCOs(N,, s), (") P. and J. Curie, Development by compression of polar electricity in hemihedral crystals with inclined faces {Bulletin de la Société de Minéralogie, t. III, p. 90; 1880). 396 LitRE XI. - DIELECTRIC BODIES. (jiii mask, outside, Feffel of the internal polarization of the crystal. Let's subject the crystal to a slight deformation. Let ()U r)U dV the components of the expansions along the axes Oç, O/}, O^ and toW àW àU dV d\ -+- ■ - t -+- - 1 'K tor, di £^C &n ■ ^f the components of the shifts. The quantities X, u, v on one side, and the quantities -~ on the other hand, will vary. We will admit the following simplifying hypothesis, which seems sufficient for the study of piezoelectric phenomena: The variations of the quantities "k, |jl, v are large compared to the variations of the quantities -!'''- Let us imagine, moreover, that the deformation imposed on the crystal leaves this crystal homogeneous; the case where the crystal would not be homogeneous would lead, in general, to considerations too complicated to develop. We will then have, inside the crystal, a uniform polarization with the following components (3) This polarization exerts the same external action as an electric layer distributed on the surface of the crystal and having density (4) a = - [al> cos(N/, 3") -h Ht, cos(N,-,j)-f- acos(N,,,3)]. Let's assume that, during the compression, the crystal has been kept isolated and that its substance is a very poor conductor of electricity. During the first moments after the compression, the X = - Tf (5n À -+- Oi, I-^ -1 ^13' '), Ilî, = - ^(^21 X -t- 02-2 !^ 023^ >), = - ^('^31 X ^032 [i. -+ 033 V) CFUP. V. - PIEZOELECTRIC CRYSTALS. 897 pressure, each element of the crystal surface retains substantially the charge it carried before the compression. If we neglect the small deformation of the crystal surface, we can say that, in the first moments after compression, this surface still carries an electric layer whose density at each point is given by equality (2). The external action of the crystal will therefore result from the action of the real electric layer whose density is given by the equality (2) and the action of the fictitious electric layer whose density is given by the equality (4) - The crystal will therefore appear covered with an electric layer whose density is !2i: =r ffo 4- 0- = (Jloo - <Â.) cos(N/, :r) - ('ift)o- lt50cos(N,-,7) H-(eo-a)cos(N,-, ^). Gradually, this apparent electrification will dissipate, as the electricity will eventually take on its equilibrium distribution at the surface of the crystal, masking the effect of the inner polarization. We will have , , , <9U , f9V , dW 0\ dr, 1)^, d\ d\\\, /à\Y à[J\. fdV O'C, dr, j and analogous forms for jjl and v. The apparent surface density S is then the sum of three similar terms, the first of which is l { L '^ç ^^j '-'s /â\Y dY\ WàU dW\ j (d\ d\}\-\ i-A~ 012 /dW dY\ /dV ()/0\ 0[ \'\ /dW d\'\ /dV à\y\ (d\ dV\ "xo-n = 0, 3=0. Following this compression, the quartz does not polarize, it does not show any apparent electrification. 2° A normal and uniform pressure is exerted on the lateral faces mn, m' n'. Let S the lateral surface mn; S' the lateral surface nn'; P the compressing weight. If A", k' are two coefficients that depend on the elasticity of quartz, we have c)U _ P dY _ P The quartz will polarize; this polarization will be determined by ties !p <âa = - TjJ l( k -f- A"' ) ç- > O ^ ' ■ 1)1) = - Tn/n(A'-r- A-')^, The face nn' will carry an apparent electrical charge Q, = -t;tZ(Ah- A')P|-. The face mm' will carry an equal apparent load and of opposite sign. The face mn will carry an apparent load (11) Qz=Tn77i{k-hk')P. CHAP. V. - THE PIKZO-ELECTRIC CBISTALS. 4o3 The face m' n' will carry an equal apparent charge of opposite sign. According to the researches of MM. P. and J. Curie, the face nn' does not manifest, under these conditions, any appreciable electrisalion. The coefficient l is therefore either equal to o or very small. According to the same research, Q2 is positive 5 therefore, the coefficient m is positive. Experience and theory agree that the load Qâ is proportional to the compressing weight and independent of the dimensions of the prism. 3" A normal and uniform pressure is exerted on the lateral faces mm' , nn'. Let P' be the compressing weight. We have f/î ' S' 0., ' S' The quartz will polarize; as l=o, from the previous experiments, this polarization will be determined by the equations X = 0, P' lll) - mm( /"- -I- /i ' ) -jTT " The bases of the prism and the side faces mm', nn' will carry no apparent load. The face mn will carry an apparent load (19.) Q; = -Tîim(Â-i-Â:')P'-g-. This load will be negative, proportional to the compressing weight and to the ratio of the non-compressed over/side area to the compressed over/side area; the proportionality coefficient has the same value rnm{k -^ k') for the two loads Q2 and Q^. These results were experimentally found by MM. P. and J. Curie. 4o4 BOOK XI. - DIELECTRIC BODIES. M. Rontgen (') has made a series of more complicated studies on the piezoelectricity of quartz. We do not want to linger longer on this study; we will refer the reader curious for further details to the Traité de Cristallographie of M. E. Mallard (2) and to the Memoirs of M. Rontgen. (') C.-W. Rontgen, JJeber die durch elektrische Kiàfte erzeugte Aenderung der Doppelbrechung des Quarzes ( Wiedemann's Annalen, t. XVIII, p. 2i3 and 534; 1882). - Ueber die thermo-, actino- und piezoelektrischen Eigenschaften des Quarzes {Ibid., t. XIX, p. 5i3; i883). - Elektrische Eigenschaften des Quarzes {Ibid., vol. XXXIX, p. 16; 1890). (") Mallard, Traité de Cristallographie, t. II, p. 555 ff. BOOK XII. DEFORMATIONS OF POLARIZED BODIES. CHAPTER ONE. THE PRESSURE INSIDE THE POLARIZED FLUIDS. § 1 - Condition of dielectric or magnetic equilibrium of a fluid compressible. We have already studied (Book IX, Chap. VIII) the conditions of equilibrium of any magnetic fluid, perfectly soft or not, but supposed to be incompressible. We will now repeat this study for any compressible fluid, magnetic or dielectric, but perfectly soft. Designating by a- the specific volume at a point of this fluid, by dv an element of volume of this fluid, we will admit that the internal thermodynamic potential of this fluid in the neutral state is of the form / <1>((T) Jr. It is to admit that the state of the fluid can be determined without having to take into account the mutual position of its various elements; it is, consequently, to neglect in this hydrostatic study the actions of capillary pressures. This is an assumption that we make not out of necessity, but only to shorten the 4o3 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. We could get rid of this hypothesis by following the methods that we have indicated elsewhere (*). We will also admit, to simplify our study, that the system does not carry any electric charge; if we did not make this assumption, we would have to take into account the various electric pressures, which we could determine as we have done in a Memoir specially devoted to this study (2). Finally, we will admit that our fluid in the neutral state is subject to two kinds of external forces: 1° Forces applied to its various elements of volume; these forces have for potential the quantity V being a function of x^ j', z finite, uniform and continvie in any point of the fluid, and whose form depends neither on the form nor on the state of the fluid; such would be the action of gravity, the action of several fixed bodies subjected to the law of universal gravitation. 2° Forces applied to the various elements of the deformable part of the surface that limits the fluid. An element <:/S of this surface bears a force P. In this formula, ^ is the magnetic potential of the system. The other letters have the meaning that we have constantly attributed to them during this Volume. This internal thermodynamic potential never appears, in any question, except by its variation; it follows that one can, without inconvenience, remove all the terms which are subject to remain constant. Let us assume the system formed by the perfectly soft fluid considered and permanent and invariable magnets. The variable part of E (V - TS) will be reduced to / *(C7)^P, the integration extending to the entire volume of the perfectly soft fluid. The variable part of the term / rT(;)lL) Jt', where the integration extends to the whole system, will reduce to the integral J .1 {DïL,!) dv, extended to the polarized fluid; we have highlighted the variable 0- which, of course, the form of the function 5 {0%) will depend on in general. Finally, if .A,, ill>, 3 are the components of the magnetization at a point of the fluid; if O is the magnetic potential function at 4o8 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. same point; if "Ç is the magnetic potential function of the external bodies, the variable part of ^ will be ^J II àx J \\\ dx dv. We can therefore, by altering only § by a constant term, write dv. (,) ,f= f^h)dv-^ fSiD\l',^)di^-^- riLl,^! dv+ fï\X^ J J '■* J H ^-^ I J W '^^ If we put (2) Y{0]-L,<7) = -r^^ , the polarization state of the fluid without coercive force will be represented by the equations x.= -Forc..)''e the work done by the external forces in an elementary modification, this condition is ex will take precedence as follows l§td^e or (4) If -^5,^0 CHAP. I. - PRESSURE OF POLARIZED FLUIDS. 409 In the particular case where the virtual deformation imposed on the fluid is reversible, that is to say where a new virtual deformation is obtained by changing all the signs of the displacements that the various points experience in the first one, the sign of inequality must obviously disappear from the previous condition, which becomes (5) oS-dîLc - o. The external work is the sum of the work d^'^ performed by the pressures borne by the surface of the fluid and the work d^"^ of the forces applied to its various volume elements: ( 6 ) fl?(r c - f^G t - disl. . We also have (7) dir.'c = V P [i< cos(P, ar) -t- ç'cos(P,7) -+- w cos(P, 2) ] '*■ If u, ç, w are the components of the displacement of the point which had initial coordinates x^y, z, we will have (8) and, therefore, ^ , /r)n ôv d"'\ , orfc - u H , eir, \ ilr i)vr f)-. / ().r à y ùz j .^ du de d\v \ d.r dy ôz \ fdW d\ ( 9 ) C?C". - OW = / - ( T- " -i- V- '' + V- <*' ) d^' J 1 \dx oy OZ I Equalities (6), (7), (9) give us the complete expression of dls)e. Let us now calculate the variation 8# undergone by the internal thermodynamic potential of the fluid. 4lO BOOK XII. - THE DEFORMATIONS OF POLISHED BODY. From equality (i), we can write (lo) oi=A + B + G, the three quantities A, B, C having the following meanings (II) A = ô / * {, 3 undergo any variations B-jI), 8\)î), 03, we know that the quantity / dx clv -U' a.> dx dv 4l2 BOOK XM. - DEFORMATIONS OF POLARIZED BODIES. undergoes a variation I\ dx OaAa. So we have c)(t') + -C)) dA. di'O ^- 't?) dT, d{'0 H- t?) dS I u (i5) L cij7 c^K ày dy dz dy J d{-0^ V ) ().Ao () fl') + -0 ) (^ "lî. (? ( t) -^ \'^ ) èia "1 '>2 J [ ûx ^/ d^ d3 ^i;. If we denote by ûfSj one of the elements of the surface S which confine to the volume V, we will easily see that (i6) C"= S Ox u cos(N/, x) ]| o?Si, N, being the normal to the surface S towards the interior of this surface. Similarly, if d^^ designates an element of the surface S contiguous to the volume V2, we have (17) G"'= olo <)(V)^-■C>■ ôx !1 u cos(N;, x) Il c^Sj. The equalities (3) give (18) (19) dx ôx ôx at ôx ôx 1 ôx ôy ôy dy dz dy f ô{V^-K^) ÔX , d(t)-f-t:)) f)iii, ^ t)(0 + -C)) 03 dz 'dz [ dD]h^ :>.V[D{1, a) dx I d;)]U -^F^JlL', a) ôy I (J;)1L2 ) dx dz dy dz ' dz According to equality (2), we have / I dDïi'^ _ a j'(;)rL,(T) _ drfiD\l,(j) ô^ l '2 F (Ole, (7 ; ôx ôx ôa dx I (J;)Il2 _ àri(D]h, J) _ à^{DVu,o. The points A,, Ao, . . . are any points located in the mass of the fluid. The volumes Q,, Qoj --- of the cavities are any zero or positive quantities. The previous equality leads to this new proposition: The quantity (It + C) cannot be negative at any point of the fluid. Let us apply this result to the points of the fluid surface itself, noting that, for these points, according to equality (28), we have P = nH-G, and we will arrive at the following conclusion: For V equilibrium of the fluid, it is necessary that any external pressure which is not equal to o is directed towards V interior of the fluid. The quantity (II + C) is never negative; it is equal to the external pressure P at any point of the surface which limits the fluid. We will designate it by the letter P and we will name it pressure at the point {x,y,z) inside the fluid. We will see 420 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. what mechanical definition this quantity is susceptible of. Let us consider, finally, any modification where the fluid does not form cavities, but where each element undergoes any variation in volume. The modification being reversible, condition (24) is transformed into an equality which can be written, by virtue of relation (28), - C V [ a ces ( N/, 07 ) + p CCS {Ni,y) -\- IV cos( N,, z )] dS -/! ^[a*(a) + a.f(^I,a)] ''^^' / du di> dw ' > \ dx dy Oz But we have ^[mcos(N,-, a7) + Pcos(N/, y) + wcos(Ni, z)]dS = -j(j^ -^ 1^ ^\ dv. If we remember that we have, by definition, P=n + G, we see that the equilibrium condition can be written /! ^[acï>(a) + .g'^(J1L,a)]- ^/:;;,. + P ^ ("£ + £ + ^^ ) ^. 1 - 0112 F(OIL, a ->^ P| 1 (au )\dx dv dw' -^Tz, from dx dw 'di But the quantity is an arbitrary function of ^, JK? ^ i ^^ quantity between braces is a continuous function of x^y. z\ then, it is easy to see that the previous equality requires that we have, at any point of the fluid mass (3o) -^[a*( *,ii . ôi ^ ' > J^ F(Jk, a-)J2 orf3IV. The second member is the sum of two terms; the first is CHAP. I. - PRESSURE OF POLARIZED FLIIDS. 423 negative for all magnetic bodies; the sign of the second one is not not known; if we neglect the quantity -~ - in front ofFi^DïL, \ , ''^'-[■dx^dJ-^-d^)'^ we find that the term (2) of the internal thermodynamic potential experiences, in the considered displacement, the following variation ch. T^ /" l 7 .^.- " / (^ff- à^ àw \ J / " "or du dv àw \ , / . ^ du ,^ dv _ dw -r- -+- llî,2 _ -t- ^2 - àx dy dz " I .^ ^ / àv dw\ _ , / dw du\ , " /du dv k 0)1,3 ( -T- + -r- 1 4- 2.1, -- + V- + =A") to, d(r)-t-'Ç) of dx dz d{X)^V) dx dx dx dy dz dz dz cos(N/, x) -h P cos(P, x) cos(N;,7) -f- P cos(P, j) cos(N,-, s) + Pcos(P, z) pX pZ V (h \ dS = 0. This equality does not have to take place whatever u, (^, w, it only has to take place if u, t^, w verify the conditions (9), so there must exist six functions of x,y, z, Ff^ F2, F3; T,, To, T3, uniform, finite and continuous in any point of the body considered, such that we have, whatever u^ r, w^ f\ d( t) + -C)) Mo d( t") ^ X')) d\\> diV") -+- -CO àB dx dx dy dx dx pX\ u d(t^-^<>) dA. d(V + X))d\\^ ()(t)^-\') o3 ^ - ! + - - - -4- p Y dx dy dy dy dz dy d(V-^'Ç) d.Ào d{XD-^V) d\i\y d(V) 4-\'>) dB - ^^ 1 _;_ -2 : u- _; ; dz (.0) s dx dz dy dCO^X")) dx -/[ Fi from dx dx dx dv dz dz cos(i/, a;) -+- P cos(P, x) cos(N/,7)-f-Pcos(P, j) cos(N/, z)-i- Pcos(P,^) dw pZ w > dv dS dw of Let's put (II) N, = F,N2=F2 N3=F3 dx dz /from the dv )] dx d(V-\-<)) dx dx II' CHAP. II. - PRESSURE OF POLARIZED SOLIDS. 433 the three quantities N,, No, N3 will be, like F, Fa, F3, three functions of x, y, z, uniform, finite and continuous inside the studied body. An integration by parts will allow to transform equality (10) into the following one: s dy ôx d^(l'^-4-V)) dzdx dx Oy ùx Oz âx Oy Oz Oyt dy âz dx ôzdy dz^ dx dy tozl dT_ dy ^.1 dz dv \ [Pcos(P, 37) - N, cos(N/, x)- T3C0S(N,-, jk)- T2Cos(N,-, ^)]" -f-[Pcos(P, j) -T3Cos(N/, a:-) - N2Cos(N/, 7)- Tj cos(Nf, ^) | r + [Pcos(P, z) - T2C0s(N/, a?) - T, cos(N,,jk) - N3 cos(l\,, ^)] tv j rfS -^ 0. This is the equality that must take place whatever u, v^ w are. We will show, first, that the coefficient of a in the triple integral is equal to o at any point inside the body considered. Let us suppose, in fact, that this coefficient is different from o at a point A {Jig- 3o) inside the body. Since this coefficient Fig. 3o. cient is a continuous function of x,y^ 5, we could draw around point A and inside the body a domain 1, at any point of which this coefficient would have the same sign as at point A. Let 2 be what remains of the body when we remove domain 1. We could impose on the body a transformation such that we would have, at any point in space 1, ii >> o, V = o, (V = o and, at any point in space 2, M = o, i^ - o, w = o. D. - II. îR 434 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. The first member of equality (12) would then be reduced to rfpX X '^'^^'^'^"''^^ ^, JHt^+<>) ^d\'0+'<)) ^N, dJ, àT,l^^_ J^ \j dx^ dx dy ^ dx dz dx dy dz \ All the elements of this integral would have the same sign and, contrary to equality (12), this integral could not be equal to o. Thus the first of the equalities is demonstrated ' ' ôx'^ ' dx dy dx dz dx /j3x; Y 4 ^'^^'^^"^'^^ Dl /'^^'^^^ ^ dH-0+\^) ^dT, ^ j ' dy dx- dy dz dx Z-X ^!i^^±^--ijl ^'^^'^-^"^'^^ _G à^V-^-i^)) ^ ^ ""^ dz dx dz dy dz'^ dx The last two are demonstrated in a similar way. These equalities make it possible to erase the triple integral in the first member of equality (12); the double integral contained in this first member must then be equal to o whatever m, v, w are, we can easily conclude that we must have, at any point of the surface of the body, / Pcos(P, ar) = Nicos(N/, rr) + T3 cos(N/, jk) -f- T2 cos(N/, z), (14) j Pcos(P,jr) = T3Cos(N/, 37) + N2Cos(N,, 7)-f-Ticos(N/, z), ( Pcos(P, z) - T2C0s(Nj, a:)-HTi cos(No jk)-+- N3 cos(Nj, z). By virtue of equalities (i3) and (i4), equality (10) is identically satisfied whatever m, v, w are. Condition (8) thus becomes, for any reversible virtual modification, dy ' dz dz dT, dy dJ f \dx ~^ toy ~^ dz J \ dx dy dz J dx ] L*^ "^ \dx '^ dy dz ) dy \ "^ [/■ .., \ dx dz j kl- - \ dx] . ;d\] toW\ - ,, ) /du tov d(v\ , (m CHAP. II. - PRESSURE OF POLARIZED SOLIDS. 435 dx dx dx dx -t-N,+ /f Let's put, to shorten, Hl^-^^--(^-^) An integral by parts will transform the previous equation into the following: IV dd\'^^ ^ d /d{] d\ d^ dx dx \dx dy dz )z )^ dx d dx d_ dy d_ dz dx H r /(^w (JU Y'-y-dx^Tz dx W J dy dx ) 1 2 ~^ " ^'^ =]l" rfc cî<^ J i ' ^/ ^ ' dy\dx ^ dy'^ dz ) ' (^K 'dy [^ '"^ t^K d{\l)-^V) k\\\A d_ ai dx c/t' 436 BOOK XII, - DEFORMATIONS OF POLARIZED BODIES. J \ dz ' dz\dx dy dz / ùz ^ ^ ,"^"^ -* M-Ni4-A-.l,2l cos(N/, ^) OX II J \ i' / SI r, .,, , , fd\j d\ dw\ !-" [d^-^-dyj-^ Ti + /.M)!,S I cos(N,, ^) ^ K^ ^ £) "^ T3 + A^i)l,.l,] cos(N,, ^) j .^S S![ , , , , . i d{] âV d\\ , ox ay dz J ' i\i. dz .,, '""+^-'> |Un.+ A3 d:p 2] cos(N,-, This equality must have a link whatever u, ^, w are. By a reasoning analogous to the one we applied to equality (12), we will prove that it is necessary and sufficient for this to have : CIIAP. II. - PRESSURE OF POLARIZED SOLIDS. i" At any point inside the polarized solid, the equality 437 d_/0[J Ox \ dx dz) dx'^ /dV dy\()j Ox / ^ dz \ Ox dz ) (i5) =-^h diXl-^O) --(T3+/.-,Ull,) Ox d_ ' ôz 4^ + /iOrL2-+-A-.i,=l (T2 + A-.l,G), \ and two other similar equalities; a" At any point on the surface of the body considered, the equality f, /d\] d\ d\ oin ^^, ^ (.6) dy -[ Nr dx dz cos(N/, z) X ^ + h 0]U -t- /crX,^ cos(Ni, x) - (T3-1- /k^I,i)1,) cos(N,-, jk) - (Ta-f" A- A3) cos(N/, z) = o, and two other similar equalities. Equalities (i 3), (i4), (i5) and (16), together with the conditions of magnetic equilibrium [Book X, Chap. IV, equations (7)], provide the conditions that are necessary and sufficient for the equilibrium of the polarized body. It remains for us to discuss the consequences of these equations. § 2 - Pressures inside a primitively isotropic, slightly deformed and polarized solid. Consider a polarized solid in equilibrium. There are six uniform, finite, continuous functions of x, y, ^, Ni, N2, N3, T" T." T3, which verify equations (i3) at any point of this solid they equations (i4) at any point of its surface. Let us draw a surface S in this solid which, either alone (y?^. 3i) or with a portion S) of the surface S (Ji^- 32), isolates a part A of the solid body. Let us denote by B what remains of this body when we have removed the part A. 438 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. Let's define, as in the previous chapter, the magnets that for part A would be equivalent to part B. Let's remove part B without changing the deformation state of part A and leaving part A subject to : i** To the action of the external forces which acted on it; 2° To the action of the magnets which acted on the primitive solid; 3° To the action of the magnets equivalent to part B. Fi g. 3i. Fig. 32. It is easy to see that the magnetization of part A remains what it was before the suppression of part B; that, on the other hand, the displacements of part A are no longer subject to the bonds imposed on them by the presence of part B; but that, in order to re-establish the equilibrium of part A, it is sufficient to apply suitably chosen pressures to the surface S; the magnitude and direction of the pressure at each point of the surface 2 are given by the equations IPcos(P, a?) = Ni cos(N,-, 3')■ Pcos(P,JK)-T3COs(N/,^) Pcos(P, s) - T, cos(N/, 3") T3COS(N,-,7)-4-T2COs(N,-, 3), N2COs(N,-,j')^-TiCos(N,s^), T, cos(N/,7)-+-N3CosrîV/, ^). In these equations, N, represents the direction of the normal to the element d^ at a point from which we want to determine the pressure, this normal being oriented towards the interior of the part A. The size and direction of the pressure are thus defined at any point of the body and for any orientation of the element passing through this point, according to the six quantities N" N^, N3, T" T2, T3. Therefore, we would say that knowing the values of these six quantities CHAP. II. - PRESSURE OF POLARIZED SOLIDS. 489 at the point {x, y, z) of the body is to know the pressures at this point. We shall not linger here to study the beautiful consequences that Cauchy and Lamé have deduced from equations (i4) - they can be found in all the Treatises on Elasticity; they are not essential to the development of the theory we are dealing with. § 3 - Fictitious pressures inside the polarized solid. Let us suppose that we give ourselves the state of magnetization of the body and the values that the six quantities take, at any point of the body N" N3, Ti, T2, T.3. The deformation of the body will be determined by the equalities (i5) and (16). If the body were not magnetized, these equalities would be of the following form dx \dx dy dz j (i5 bis) ilj. ^[; dz\dx to )] (i66fs) dy \ày dx \ dx dy dz ■dU dV dW\ dUl ,^, , \dx = - [Ni cos(N/, x) H- T3 cos(N/, y) 4- T2 cos(N,-, z)] xi ^^ ^jcos(N/,^)H-[i(|-^4- j^jcos{^i,z) These equations and the analogous equations deduced from them by rotating permutations determine the deformations of the unpolarized solid as a function of pressures; they play the same role, in the study of the deformation of elastic and primitively isotropic solids, as the compressibility equation does in the study of fluids. If we compare equalities (i5) and (16) with equalities (i5 bis) and (16 bis), we arrive at the following conclusions: We can, in a polarized solid, give the equations that 44o BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. link the deformations to the pressures in the form they have for a non-polarized solid, but with the condition of replacing the real values of the six quantities N" N2, N3, Ti, T2, T3, by fictitious values N' N' N' T' T' T' 11 j, i.y^, l ilj, Ij, ±2) -'-35 linked to the former by the relationships (17) N'.-N3 + dx dCxo + t?) dx dx t; = Ti H- A "i)i>e, T;j = T3-hAol,alL. Equalities (17) can be transformed. If we refer to the laws of magnetic equilibrium [Book X, Cliap. IV, equations (7)], we see that we have CAD dx 2<^. The equalities (17) thus become (18) N'i = Ni - il + AD1L2 + A-.1.2, n; = N2 - >> -i- h^\U -t- A 1(1,2, N'3 = N3 - J; + h D1L2 + k 32, T;=:Ti + A'iiï,a, t; 3-^3 A-..\,D1,. If we refer to the expression of d/. -f-2/î Hv^r ^^)---(^^ -1^-) -11!, 2+ -^- G2 dj' àz \dy '^ dx / "" "]. (19) CHAP. H. - PRESSURE OF POLARIZED SOLIDS. 44 I and if we notice that the quantities are very small, we can make the equations (i8) undergo a new simplification and write them / N; = Ni - (/- /t)0rL2-f-A:.lo2, N'2 = N2- (/- h)D]U-\- /rDb2, N;-. N3- (/- h)DïL^-i-kQ\ T'i = TiH-/nlî,G, T'2 = T,-i-/.SJlo, An interesting case is when we look at the two quantities h and k as very small before/. In this case, if one refers to the equations of magnetic equilibrium [Book X, Chap. IV, equations (7)], one sees that a deformation of the first order imposed on the solid only modifies the magnetization coefficients by quantities of the second order. In this case, equations (19) reduce to (20) t; ^ T, If we then refer to equations (i4)? q^^i define the pressure exerted on any element dS passing through the point (.37, r, z), we see that the substitution of quantities N' N' N' T' T' T' given by equalities (20) to quantities N" N2, N3, T" T" T3, is equivalent to the addition of a fictitious pressure, normal to the element dSj to the pressure that this element actually supports. This pressure, independent of the orientation of the element, has the value (21) n^-forL^dS. This pressure is negative, because the quantity / is positive; - -r is, in fact, the magnetization coefficient of the non-deformed body. The additional Jictive pressure is negative at any point 442 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. of the polarized body; it is equal to the quotient of the square of V magnetization intensity by twice the magnetization coefficient of the undeformed body. When we neglect in front of /the quantities h and /:, and, a fortiori, the quantities h --, - ..., A:--, - .., ox ox we have - = ^ ! [^i^T- r-'^T- r-'^^r f and the equality (21) can still be written If the body is not very magnetic, O is negligible compared to ') dx ^\o dr2 llî, d-i(V-hX>) dx ôy d ( 10 -f- -C ) d\. d(V)-^t)) dlll, at ( V) -t- V) ) of âx Ox <)z dx at dx dx The conditions of magnetic equilibrium [Book X, Chap. IV, equalities (y)] then give <)(t)-(--C>^ d.\. d{ V) -I- V) ) f)\l dx"^ dx at dx d{X)-\--*i)) dZ^ dz dx -_[/- /^ ^ àWV\ dPïi ~~ \y \dx dy dz / \ dx _ /àV W d\ d_\^ dW ^32 \dx dx dy dx dz dx [ \ (^2 ~^ dy ) dx \ dz ) dx \ dy dx / dx ] dx dv\dx ày ~^ dz j -H k ( .1,2 -- - -4- 1)1,2 ^_ 3-2 .^ dx^ dx dy dxdz d ld\\} dxj\ dx According to these and other similar equalities, the equalities (22) will become (23) _a_/aU dY\ d /dW dV\ ~^^ày\àf ^ dx) '^^di V"^ '^ dz ) y ^^ dx\àz ' dy j ^' " dx\dx ~^ dz j " '' dx \dy dx) J h dx l'dX^- drX\\^ d,l3\ " and two other analogues. If we eliminate the six quantities N" N2, Na, T" T,, T3 i-^Â)' 444 BOOK XH. - THE DEFOUM.VTIONS OF POLARIZED BODIES. between equations (i4) and (i6), remembering that we obtain the following equations, which must be verified at any point on the surface of the solid, rxf^" + i?v + ^)H-"f + + .. "ar..+ /t-.A,=lcos(N,.x) -I- \'^(j^--^f^') +^--^S 1 cos(N/, ^^ + Pcos(P, a7)=o and two other analogues. The equations (aS) represent the second order partial differential equations that the displacements U, V, W must verify and the equations (24), the boundary conditions that these same displacements verify. The integration of all these equations and the equations of the magnetic equilibrium will solve the general problem of the deformation of a polarizable solid placed in determined conditions. This problem will, in general, present insurmountable difficulties. However, it is possible to solve it in a case which, although peculiar, is of great interest and throws a great light on magnetic actions. Consider a body which, in its natural state, is isotropic and is not subject to any external force or pressure. This body is splerical. We place it in a uniform magnetic field. We will demonstrate that it expands in the same way at all its points so as to transform itself into an ellipsoid of revolution whose axis of revolution is directed along the lines of force of the field. Let us prove, indeed, that there is a similar deformation satisfying the conditions of magnetic equilibrium and equations (23) and (24). First, the ellipsoid is homogeneous; its axes do not differ from each other by quantities of the order of -; its coefficients CIIAP. II. ~ PRESSURE OF POLARIZED SOLIDS. 445 of magnetization also differ from each other only by quantities of the order of j-- The quantities Jl,, ^i^), S vary, from one point to another of the body, only by quantities of m To the order of ( t- ) ' as can be easily seen. As we neglect quantities of this order, the magnetization of our body should be considered as uniform. By reason of symmetry it will coincide with the axis of revolution of the ellipsoid which is directed along the lines of force of the field. Let's place the x-axis along the direction of the magnetization. We will then have (25) 'dz = 0, cJl = dJL; ùY dW dy dz d\] toW dz dx dV dx dy The magnetization and the deformation being uniform, the external forces being zero, equations (28) are identically satisfied. The equations (24) are reduced, by means of equations (26), to (26) r X -^ 2 a - (A -+- A-) Oit 2] '-^ -f- 2 ( À - h 0112) '^ +(/_(_ A + /c) 01L2 = o, d\] d\ X_(A + /,)01L2] _±i _^2(Xh-[x - A01L2) - 4-(/ + A)orL2 These equations (26) can be satisfied by posing k{^x) - (f+h)]x - kh ;)1L 2 (27) Ox d\_ dy \>.[ik -H 2fJL - (J -H ArpK^J Î)\U, dW 'dz A-X - 2 'i(f^ h) - k(h^ /OOIL^ 5 2[jl[3X-+-2[x - (3A + A)0IL2J *" These equalities (27) define the deformation of the sphere. In the particular case, already studied in the previous paragraph, where we neglect the quantities h and k in front of the quantities \ and/, we find (28) d\^ _dY_ dx dy dz 3X 2[X 446 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. The quantity "=^ is the magnetization coefficient of the body in its natural state. The quantity G = 3X -j-li is the cubic compressibility coefficient of the body. The formula (28) allows us to write ^"^'^^ âx df ~^ (Jz"~ ~Yc~ ' If we neglect the quantities h and /:, a magnetic sphere^ placed in a uniform magnetic field, expands while remaining similar to itself. The cubic expansion is obtained by multiplying the square of the magnetization intensity by the cubic compressibility coefficient of the body and dividing by twice the magnetization coefficient. If we denote by J the intensity of the field, we have [Book VIII, Chap. II, equality (7)] and the equality (29) can also be written to dW G cr (3o) dx dy dz it 4 These results can be extended to a body of any shape placed in a uniform magnetic field, provided that this body is weakly magnetic and that the quantities h and k are neglected before y. This body will expand uniformly in all directions, and the cubic expansion will have the value au dy dW GcJ2 ox of toz 2 CIIAP. II. - PRESSURE OF POLARIZED SOLIDS. 447 g 5. - Comparison of the results of the theory with the results of the experiment. The preceding theory applies equally to dielectric and magnetic bodies. However, when it is applied to a dielectric body, care must be taken to ensure that this body does not carry any electric charge and that its virtual deformations are not related to displacements of the electrified bodies; this restriction would not be achieved by a dielectric plate to which the two armatures of a capacitor would be stuck. The expansion of a dielectric placed in an electric field has been known for a long time. Abbot Fontana (*) had already observed that the volume of a capacitor increases when it is charged. This discovery had fallen into oblivion when the fact she had observed was reported again by M. Govi (-), then studied successively by M. Duter (^), by M. Righi (*) and by M. Quincke (s). Let us imagine a dielectric plate placed between the two plates of a capacitor, and not fixed to these plates, a condition rarely realized in experiments, and whose forgetfulness is perhaps the cause of the divergences that exist between their results. If we denote by V and V the potential levels of the two plates, by A their distance, according to the formula (3i), the cubic dilatation of the dielectric plate has the value au av ^_ Q (V- v)^ dx dy dz A^ It is proportional to the square of the level difference (' ) Quoted in a letter of Volta to professor Landriani {Lettere inédite de Alessandro Volta, printed in Pesaro in i83i, p. i5 et sqq.). (") Govi, Nuovo Cimento, t. XXI and XXII. - Reports, vol. LXXXVII, p. 857; 1878. (') Duter, De la dilatation électrique des armatures des bouteilles de Leyde ( Comptes rendus, t. LXXXVIII, p. 1260; 1879). (* ) RiOHi, Sur la dilatation du verre des condensateurs pendant la charge {Comptes rendus, t. LXXXVIII, p. 1262; 1879). (' ) Quincke, Ueber elektrische Ausdehnung (Monatsberichte der Akademie der Wiss. zu Berlin, p. 200; 1880). 448 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. potential of the two armatures and in inverse proportion to the square of their distance. This is the result found by M. Quincke. Mr. Duter and Mr. Righi observed a dilatation proportional to the inverse of the simple distance. In the experiments of these physicists, the insulating blade was not independent of the armatures, which perhaps explains this discrepancy. CHAP. 111 - MAXWELL'S THEORY. 4^9 CHAPTER III. MAXWELL'S THEORY. § 1 - History of the theory of pressures inside polarized bodies. The first author to try to specify the nature of the pressures inside a polarized body was Maxwell. The most important propositions of his theory were published in i86i-i86a in the Philosopliical Magazine (') and in i865 in the Philosophical Transactions (-). This theory was later developed in his Treatise on Electricity and Magnetism, the first edition of which appeared in iB'jS (^). Maxwell's theory has been welcomed by physicists as a work of genius; it is regarded as one of the most important theories of modern physics. The propositions that constitute this theory can be reduced to the following two: 1° If, inside a polarized medium, solid or fluid, we trace a surface separating one part of this medium and we remove the bonds resulting from the presence of the other part, we will re-establish equilibrium by applying pressures to this surface. If we designate by N< the normal to this surface at a point (x, j^, 5), this normal being directed towards the interior of the portion of body which it limits, the magnitude and direction of the pressure at (' ) J.-C. Maxwell, On electromagnetic field. (') J.-C. Maxwell, On electromagnetic field {Philosophical Transactions of the royal Society of London; i865). (' ) V" Part, Chap. V; IV Part, Chap, XI. D. - II. ag 45o BOOK Xll. - THE DEFORMATIONS OF POLARIZED BODY points {x^y, z) are given by the equations / P cos(P, x) =-- Ni cos(IV,, x) + T3 cos(Ni. j) -f- T2 cos(N,-, ^), (1) Pcos(P,j)=:3T3COs(N,-, a7)-f-N2Cos(N,-,7)+Ticos(N;,3), ( P cos(P, ^) = T, cos(N;, x) + Ti cos(N/,7) -+- N3 cos(Ni, z), equations in which N,, N2, N3, Ti, Ta, T3 are six functions of (^,y, z) defined by the following equalities (2) ^ Ti = f - + /t^ "^^ ^'^ "^'^ ) (^( O + -^'^ ) N, N3 = 4'rr / c^K toz ^ ^ , _i_ \ ()(" -f- -c^) ()(X!) + 1:)) 4'!i / dx dy In these equations, k is the magnetization coefficient of the substance, which is assumed to be invariant. Moreover we have, according to the notation we have adopted, 2" The deformations of the polarized medium are related to the six quantities N,, No, N3, T" T2, T3, by the relations established by the theory of elasticity for the case where the medium is not polarized. To imagine more easily the laws which, according to Maxwell, govern the pressures inside a polarized solid, let us take CHAP. III. - MAXWELL'S THEORY. 45l the j' axis and the ;; axis tangent to the level surface which passes through the point (x^y^z). We will then have - d]r-=''' - di - = *^' and the equalities (2) will become Ti= T2= T3= o. We can see from equations (i) that a surface element normal to the line of force passing through a point of the body bears a normal pressure which has the value at each of its points while an element parallel to the line of force supports a normal traction whose absolute value at each point is equal to the value of the previous pressure. The deformations being linked to the pressures, within the polarized medium, in the same way as within the medium subtracted from the action of any field, we see that the body experiences, at each point, a contraction in the direction of the lines of force and a dilation, equal in absolute value to this contraction, in any direction normal to the lines of force. These pressures and tensions do not become equal to o at the same time as the coefficient A". If we place in a magnetic field a body which is not susceptible to magnetization, any superficial element taken inside this body, normally to the lines of force of the field, supports a normal pressure which has for value at any point JL /^V iTz\dx] (3) 452 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. while any element parallel to the lines of force supports a normal tension which has, at any point, the same absolute value as the previous pressure. Maxwell had assumed the value of the magnetization coefficient k to be independent of the deformations undergone by the body. M. H. von Helmholtz (') noticed that, in fluids, this coefficient should be regarded as being, at each point, a function A"(." o 2/ If we assume k = o, we also have K"= o. Assuming, in addition, h = o, we find K'=: o. Mr. Lorberg and Mr. G. Kirchhof found that in a polarized elastic solid, Maxwell's equations (2) had to be replaced by more complicated equations. Using the notations we have just introduced, these equations can be written as follows: (6) N, ^-u^--") K"\ d(V-{-V) d(ll-^\^) dy dz 4tt % J dz dx ^ T,= (-L 4.K 1 ) dx dy CIUP. III. - MAXWELL'S THEORY. 455 These equations are similar to those of M. H. von Helmholtz, if we make K =o and Maxwell's, if we assume both K" = o, K' --^ o, that is (7) k - o, Il - o. This theory, founded by Maxwell, developed by M. H. von Helmholtz, M. E. Lorberg and G. Rirchhoff, has been, as we have said, adopted by almost all physicists. It is not, however, without serious causes for doubt. The known laws of hydrostatics teach that at a point in a fluid, the pressure is normal to the element led by that point and independent in magnitude of the orientation of the element. According to Maxwell, this would not be the case inside a polarized fluid: the pressure exerted on an element normal to the lines of force would change into tension for the elements parallel to the lines of force. This difficulty had not escaped Maxwell. "The hypothesis, he says (' ), of such a state of tension existing within a fluid dielectric, such as air or turpentine, may appear, at first sight, to be in contradiction with the established principle, that at one point of a fluid the pressures are equal in all directions. But, when we establish this principle by considering the mobility and equilibrium of the parts of the fluid, we admit precisely that there exists in the fluid no action of the kind we suppose to occur along the lines of force. The state of tension which we have just studied is perfectly compatible with the mobility and equilibrium of a fluid; for we have seen that, (' ) J.-Cr.. Maxwell, Traité d'Électricité et de Magnétisme, translated into French by G. Seligmann-Lui, t. I, p. 178. 456 BOOK XII. - DEFORMATIONS OF POLARIZED BLOWS. if a portion of the fluid has no electric charge, it is not subject to any resultant force due to the tensions on the surface, however intense these tensions may be. It is only when a portion of the fluid is charged that its state of equilibrium is disturbed by the tensions acting on its surface, and we know that, in this case, it tends efifectively to move; therefore the assumed state of tension is not incompatible with the equilibrium of a fluid dielectric. " The difficulties of Maxwell's theory have been deeply studied by M. Brillouin (' ) and by M. E. Beltrami (2), without these authors having succeeded in overcoming them. E. Mathieu (^) also demonstrated that Maxwell's theory could not agree with the principles of the theory of elasticity. Finally, in the beautiful work he recently devoted to the exposition of Maxwell's ideas, M. Poincaré (^) insists again on the numerous paradoxes raised by the theory of elastic actions within dielectrics. All the eminent authors we have just quoted, while pointing out the numerous difficulties of Maxwell's theory, regard these difficulties as paradoxes which will one day be explained, and continue to believe in the truth of this theory. None of them goes so far as to consider as inaccurate the expression, given by the English physicist, of the pressures within a polarized medium. However, this is the conclusion we will come to. In the two previous chapters, we have resumed the study of polarized bodies, solid or fluid, by trying to put in place (' ) M. Brillouin, Essai sur les lois d'élasticité d'un milieu capable of transmitting actions in inverse proportion to the square of the distance {Annales de l'École Normale supérieure, 3^ série, t. IV, p. 201; 1887). ('^) E. Beltrami^ Sull' interpretazzione meccanica délie formole di Maxwell {Memorie délia R. Accademia delie Scienze delV Instituto di Bologna, session of February 1886). (') E. Mathieu, Theory of Potential. Second Part: Electrostatics and Magnetism, p. 110. (*) H. Poincaré, Electricity and Optics. I: Maxwell's Theories and the Electromagnetic Theory of Light, p. 88. CHAP. III. - MAXWELL'S THEORY /{5y a complete rigor, to dispel all doubts. We have recognized : 1° That the pressures inside a similar medium are not determined by the laws given by Maxwell, even when we admit the approximations adopted by Maxwell; 2" That the deformations of the medium are not related to the six functions on which the pressures depend by the laws which express this same relation within a non-polarized solid. Thus, for us, the difficulties presented by Maxwell's theory are not paradoxes that must sooner or later find their explanation, but contradictions that expose its inaccuracy and must make it be rejected. In the first rank of these contradictions, let us quote that which consists in admitting, within a tluid, the existence of a pressure which is not normal to the element on which it acts, nor independent of the orientation of this element. Experience, moreover, leaves us at ease in rejecting Maxwell's theory. It has been found, in fact, that polarized bodies expand in directions normal to the lines of force, which is consistent with both our theory and Maxwell's; but it has never found the contraction in the direction of the lines of force predicted by the theory we reject; M. Quincke (' ) has even found that a polarized dielectric expands uniformly in all directions. It is true that his method, as pointed out by M. J. Curie (2), is not free of criticism. We shall therefore reject Maxwell's theory completely, and retain only the theory, in conformity with the principles of Hydrostatics and Elasticity, which we have developed in the two preceding Chapters. § 2 - Cause of the inaccuracy of Maxwell's theory. Maxwell's theory, however erroneous it may seem to us, has been (' ) Quincke, Ueber electrische Ausdehnung {Monatsber. der Berl. Akad. der Wiss, p. 200; 1880). (") See PoiNCARÉ, Électricité et Optique, t. I, p. 29^. 458 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. adopted and admired by such eminent physicists, that we do not wish to be satisfied with this dismissal; we shall expose here the method which serves to demonstrate the formulas of this theory in order to mark the precise point where Terror slips into the sequence of reasoning. The presentation we are about to give is very close to that of M. H. von Helmholtz. Like this illustrious physicist, we shall confine ourselves to the study of polarized fluids; the remarks we shall make would easily be extended to solids. The presentation we are about to give will not only serve to highlight the erroneous point of Maxwell's theory. It will also have the advantage of introducing us to several interesting formulas. Let us consider any magnetic fluid, placed under the action of permanent magnets and on which the magnetic equilibrium is established. Let us give this fluid any infinitely small modification, consisting of an infinitely small change of shape and an infinitely small change of magnetization. This modification can be decomposed into two others: i" a change of magnetization without change of shape; 2° a change of shape without change of magnetization. Since magnetic equilibrium is assumed to be established on the fluid, no uncompensated work is done in the first modification. The uncompensated work done in the total modification is therefore reduced to the uncompensated work done in the second partial modification. This uncompensated work can be seen as the work of the actions that tend to deform the fluid; if it is equal to o, the fluid is in equilibrium. This uncompensated work consists of two parts. One would remain unaltered if the same modification were imposed on the same fluid previously returned to the neutral state; the other would be cancelled out under these conditions. Let us denote the latter by dx and calculate it. If we refer to the notations used in the previous Chapters [Chap. I, equalities (i i) and (i4)]j we have rfT = - (B + G'+G"-f-G"') CHAP. ni. - MAXWELL'S THEORY. OR [Chap. I, equalities (i3), (i5), (16), (17)], 45.J di ') ().f(,On, a)1 /du dv dw . ' ' -^ ^- ] dv '■m (8) -/p(01L, J { V ^^ àx dy dx dz dx J r cJCD-f-t?) dX ()(X')-t--0) d^ d(XD-+--Ç) d31 ^ L dx dy dy dy dz dy J dy dy dz &io rfa? dz d{t-)-^-Ç) dx dy u cos(N/, x) dz rfS. dv Let's put (9) (10) X = Y = dx 4r -^ c^^ It ) -^ V^ ) dcT Dl (J.r dy d2( !')-(-'(?) dx dz ^ Dî, Dl -f and equality (8) becomes ( dx = ^ P[mcos(N;, a?) + Pcos(Ni, jk)-+- (vcos(N/5)) t/S -h f{Xu-hYv^Zw)dv. If we refer to equalities (18), (19) and (ao) of Chap. I, we can write equalities (9) and (10) in the following form (y bis) (10 bis) P = ^{D, a) - ;)rt2 (JJ(,m,a) ~ dx \_ to), These expressions can be put in another form. Consider the equality M, My à_B __ \(\^-h<)) dx dj^ dz 4''^ By virtue of the equalities (the), it becomes ^j[>-^4-A-(a)]- ^^- ^ + _j[,-^4TrA(a)]^ (="- Let's multiply the two members of this equality by - -r and we will obtain the following equality ô-z\ [î^.-^''^'^\^ ô-z {' With this equality, the expression of X, given by the first 2 2 (3) 462 MVRE XII. - DEFORMATIONS OF POLARIZED BODIES. first of the equalities (12), becomes dx [f,-^-(')][^^^^'T (i5) ày\_^'K'^ \ dx dy \ The expressions of Y and Z can be put in an analogous form. Let's put L4i^ J àz dx dx dy and equality (i5) becomes (Ib) A - h -r 1- -3-. dx dy dz The quantities Y and Z have analogous expressions. The uncompensated work of the magnetic actions that tend to deform the fluid can therefore be written, according to the equalities (i i), (i3) and (16), rfx = - g h{<7) -+- a ^^^n n(/0 + 1?) Il M cos(N;, x) Il dS (-7) /[(t dy -r:; + dy ^ dz)' \ dx dy dz ]di^. CHAP. III. - MAXWELL'S THEORY. 463 We have assumed the continuous magnetic fluid. We could have assumed it to be formed by two distinct fluids, separated by a surface of discontinuity S. Let us denote by N/ the normal towards the interior of the first fluid to one of the surfaces S or S which limit the volume ç of the first fluid and by N^' the normal towards the interior of the second fluid to one of the surfaces S' or S which limit the volume v' of the second fluid. Let us designate by N' N' N' T' T' T' what happens to the quantities N" N2, N3, T,, T" T3 when we replace the quantities <7 and A-(o-) relative to the first fluid by the quantities a-' and k() It "'cos(n;-, x) 1! { rfs. A-'(a') o?a' This equality is due to M. von Helmholtz (*). Let us imagine, in particular, that it is a magnetic body immersed in an unlimited medium which is itself magnetic. The surface designated by S vanishes. The surface S becomes the surface of the body itself. The integral relative to the surface S', rejected at infinity, is equal to o, if, at infinity, (O -+--) cos(N,, X). 2 L4^ aj J If we compare this term, coefficient of ud^ in the second member of equality (20), to the coefficient of the same quantity a û?S in the second member of equality (20), we can see that the coefficient of ud^ in the second member of equality (20) is the same as the coefficient of the same quantity in the second member of equality (20). < ■ ) E. LoRBERQ, Ueber E lektrostriction ( Wiedemann's Annalen, vol. XXI, p. 3 1 '(; 1884). D. - II. -*" 466 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. cond member of equality (19), we find that the latter outperforms the former by - Ï4- -^'(a')- a' "LKip.] n'(XD + \'^) cos(N,-, ce) 2 L4TI a^ 1 Equality (7) of Chapter II thus takes the following form here: dS (7) G = V (9~-2(J<) u cos(N,-, x) "I'm not going to do it.] \ "I'm not going to do it, / to V \ dB + [(/'o + />,g-^...)^V...]j "rf.+.... Let's write that A 4- B 4- G - d^e = o, after transforming the expressions (4) and (5) of A and B to t w aï5 470 BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. mojen of integrations by parts. We will have, all reduction done, -1- r. . 1 cos(N,■,7)-^ r.. .1 cos(N/, z) -Pcos(P,ir)| u dS (8) (_J[pX + a./^-f-...+ ;'^(cp-^4.) -+- - (/iJU-K miil'o -h 3 + /?,^l,2 + . . .) \ - / . . . K' rft^ - / . . . (V (/t^ = 0. This equality must take place whatever w, ç, w. By reasoning analogous to that which we did in Chapter IT, we will find that we must have : i" At any point of the crystalline medium, three equations of which the first is (9) = - pX- (Zi .^l, _|_ ,?ijx)<,-+- n,3-i-/)icAD2 -I-. . .) 2" At any point on the surface of the crystal, three equations of which the first is l U^ - "11 -^ - !-- - - cos(N/, x)-h\ ... cos{Ni,y)-+- - - - cos(N,,5) { +(..-) cos(N/, y) + (. . .) cos(N,-, ^) h- P cos(P, x). Equations (9) are the second-order partial differential equations that must be verified at any point of the crystal by the displacements U, Y, W; equations (10) are the conditions that these displacements must verify at the crystal boundaries. CHAP. IV. - LKS ELECTRICAL DEFORMATIONS OF CRYSTALS. 47 1 The general integration of these equations would lead to the solution of the following problem: Find the deformation that a dielectric crystal undergoes when placed in an electric field. We will approach this problem only in an extremely particular case. We will suppose that the field is uniform; we will suppose, moreover, that the crystal is very weakly dielectric, so that, in our equations, we have only to preserve the terms of the first degree with regard to the quantities U, V, W, A,, garlic,, 3. Finally, we will assume that no external force acts on the crystal. It is easy to see that if, under these conditions, we assume uniform deformations inside the crystal, as well as polarization, the equations (g) are identically satisfied. The first of the equations (lo) will become ( II ^^ -4-. .. j cos(N/, a')-i-(...)cos(N,-,jK) + (.-.)cos(N/, 2) = [(^0 - li)cA9 -h {rriQ - /ni) -+- ("o - ^i) 3]cos(N/, x) - [/2'^-i- m^^W) -4-"2 3] cos(N/,7) - [ /s cil) H- ms l)î) -1- na 3 ] these ( Nj, xî ). We have assumed that electrical equilibrium is established on the crystal; we have further assumed that the crystal contains no free electricity; these two assumptions are compatible only if the crystal is not naturally pyroelectric. We must therefore assume /fl - o, mo - o, "0 = 0, so that the previous equation becomes the first of the equations 'au ^ +■ - -) cos(N/, x) -h (...) cos(N/,^)-t- (. . .) cos(N/, z) - (li,Ào -f-m,l)b -I-"i3)cos(Ni, 37) 1 -{liX-hm2^i\)-hni€■)cos{Ni,y) - ( /s "As -4- ms 1)1) -(- /i3 3)cos(N/, z), These three equations will be verified at any point on the surface 47^ BOOK XII. - DEFORMATIONS OF POLARIZED BODIES. which limits the crystal, if we can equalize separately, in each of them, the coefficients of cos(N,-, x), cos(N,-, jk), cos(N/, z). We thus obtain nine equations , , l year 1- . . . = /isAf) -+- millî) M- /ii3, (12) ^ Ox which reduce to six; so that the six deformations are determined, as a function of JU, a)i). G, by six linear equations. Two cases are to be distinguished: i" The body is neither pyroelectric nor piezoelectric. We have then l\ = o, ^2 = 0, . . . , The equations (12) become They give d\] dY d\ dY d\Y d\Y dU toY oz ay ôx dz ' dy dx If we place a weakly dielectric crystal, which is neither pyroelectric nor piezoelectric, in a uniform field, the deformations that this crystal experiences are of a higher order of magnitude than the polarization that it takes. In general, they will be inaccessible to the experiment. 2° The body is piezoelectric. In this case, the quantities hi hi - - - , are not all equal to zero. Equations (12) determine the six deformations as linear and homogeneous functions of the com CHAP. IV. - ELECTRICAL DEFORMATIONS OF CRYSTALS. 473 of the polarization; therefore, also in linear and homogeneous functions of the field components. If we place a weakly dielectric and piezoelectric crystal in a uniform field, the deformations that this crystal experiences are of the same order of magnitude as the polarization that it takes. These deformations vary proportionally to the intensity of the field in which the crystal is placed. They change direction with the field. These phenomena have been observed by experiment. When MM. P. and J. Curie had demonstrated the possibility of electrifying certain crystals by compression, M. G. Lippmann (' ) announced, as a consequence of the principles of Thermodynamics, that the dimensions of these crystals should vary when the crystals were placed in an electric field. Shortly afterwards, MM. P. and J. Curie (-) recognized by experiment the fact announced by M. Lippmann: a quartz plate, normally cut to a hemihedral axis, and placed between the two plates of a condenser, undergoes a dilatation or a contraction; the dilatation is proportional to the difference in potential level that exists between the two plates and in inverse reason of their distance. By optical methods, M. Rôntgen (-') and M. Kundl (') verified the fact discovered by MM. P. and J. Curie. We will stop here the study of the electric deformations of crystals. We have marked the method by which the entirely general laws of this deformation might be obtained. The present sketch could thus serve as a starting point for a complete and extensive theory of this class of phenomena. We (') G. Lippmann, Principle of the Conservation of Electricity {Annales de Chimie et de Physique, 5" série, t. XXIV, p. i45; 1881). (") P. et J. Curie, Déformations électriques du quartz {Comptes rendus, t. XCV, p. 914; 1882). (') RôNTGEN, Ueber die durch electrische Kràfte erzeugte Aenderung der Doppellbrechung des Quarzes {Wiedemann's Annalen, t. XVIII, p. 2i3 et 534; i885). (*) KuNDT, Ueber das optische Verhalten des Quarzes ini elektrischen Felde {Wiedemann's Annalen, t. XVIII, p. 228; i883). 474 BOOK XII. - LKS DEFORMATIONS OF POLARIZED BODIES. We will not develop this theory, as it would make the present volume too long; what we have said is sufficient to show how the phenomenon of the electric dilatation of crystals takes its place in the perfectly ordered table of magnetic and dielectric phenomena which is traced for us by the Thermodvnamiquc. END OF VOLUME TWO. TABLE OF CONTENTS FROM VOLUME II. BOOK VIL Magnetic forces. the ages Chapter I. - First keystones i § 1 - Magic poles i § 2 - Magnetic elements. Action of a magnet on a pole.... 4 § 3 Mutual potential of two magnets 8 § 4 Forces acting on a rigid magnet . 1 1 § 5 - Actions of distant magnets. Magnetic moment 12 Chapter II. - Determination of the law of magnetic actions and of the intensity of the earth's magnetism 14 § 1 - Definition of the elements of terrestrial magnetism i4 § '2. - Determination of MH 17 § 3 - Determination of ^ 26 Chapter III. - The magnetic potential function and the magnetic potential ^^ § 1 - The magnetic pole function outside a magnet... 35 § 2 The magnetic potential function inside a magnet . 87 § 3 - Magnetic potential 44 Chapter IV. - The equivalent dummy distributions to a 5i magnet § 1 - Fictitious distributions equivalent to a 5i magnet § 2 The surface distribution equivalent to a magnet and the derived Lejeune-Dirichlet problem 56 § 3 - Experimental methods for the study of the fictitious distribution. 60 Chapter V. - The Lejeune-Dirichlet derivative problem {j:,y, z) ,8a § 4 - Boundary conditions satisfied by the function \^^(a:,jK, z). i83 § 5 - Poisson's Apocalypse 186 Chapter III. - The problem of magnetization by influence admits a and only one solution 188 § 1 - Existence of a solution 188 § 2 - For magnetic bodies, there is only one solution to the problem of magnetization by influence. It corresponds to a stable magnetization 190 Chapter IV. - Some Theorems on the Magnetization of Magnetic Bodies 197 § 1 - Perfectly soft bodies have no magnetism residual 197 § 2 - Bodies that magnetize uniformly in a uniform field. 198 § 3 - Determination of the magnetizing function. Saturation. 2o3 Chapter V. - Equilibrium and motion of a magnetic mass in the presence of permanent magnets 207 § 1 - General equation of motion of a perfectly soft body § 2 - Instability of the equilibrium of a magnetic body in the presence of of permanent magnets. 2i5 Chapter VI. - Impossibility of diamagnetic bodies 221 Chapter VII. - Magnetization of a magnetic body in a magnetic medium 228 § 1 - History 228 § 2 - Magnetization of a soft body immersed in a perfectly soft medium 23o Chapter VIII. - Pressure of an incompressible magnetized fluid 235 § 1 - How, in an incompressible magnetized fluid, are arranged magnetic elements 235 § 2 - Pressure exerted by a magnetized fluid 247 § 3 - Experience of M. P. Joubin 249 § 4 - Shape of the separating surface of two magnetized fluids 252 478 TABLE OF CONTENTS. Chapter IX. - Actions exerted on weakly magnetic bodies aSG § 1 - Basic formula aSG § 2 - Theorem of Mr. Edmond Beequei'cl 269 § 3 - Faraday's law 260 § 4 - Instability of the equilibrium of a low-magnetic body 263 § 5 - Does Faraday's law extend to small strongly magnetic bodies? Criticism of the pull-out method 26 CiiAPiTRE X. - Magnetic spectra 271 § I. - Spectra formed by weakly magnetic powders 271 § 2 - Spectra formed by strongly magnetic powders 278 Chapter XI. - Heat and magnetization 2-8 § 1 - Heat involved in the displacement of a magnetic mass 27H § 2 - Specific heat of a magnetized body 28) BOOK X. The magnetization of crystallized bodies. Chapter I. - Magnetic equilibrium equations on crystallized bodies 289 § 1 History 289 § 2 - Magnetic induction surface 29.) § 3 - Existence of a state of equilibrium 296 § 4 Equations of magnetic equilibrium 297 §5 Determination of the distribution that is appropriate for the 3oo equilibrium § 6 Remark on homogeneous hemimorphic bodies Sol Chapter II. - The Poisson SoG theory § 1 . - The two magnetic induction surfaces 3o() § 2 The determination of the magnetization reduced to the integration of a partial differential equation 309 § 3 - Stability of the magnetic equilibrium 3i t Chapter III. - Action of a uniform magnetic field on a body crystallized 3i3 § 1 - Magnetization of a sphere or a crystalline ellipsoid in a uniform magnetic field 3i3 § 2 - Forces which solicit a crystalline sphere in a uniform magnetic field. : 3iC § 3 - Experimental checks 821 § 4 - Action of a uniform field on a weakly magnetic crystal immersed in a weakly magnetic medium 325 TABLK DKS MATERIALS. 479 aife" Chapter IV. - Magnetization of slightly deformed bodies 33 1 § 1 - Magnetization of any slightly deformed body 33'| § 2 - Magnetization of a slightly deformed isotropic body . 335 BOOK XI. Dielectric bodies. Chapter I. - Thermodynamic potential of a system containing dielectrics 339 § 1 - Thermodynamic potential of a system containing electrified and polarized dielectric bodies 331) § 2 - Dielectric and pyroelectric bodies 344 § 3 - Transformations of the potential of a system containing dielectrics 345 Chapter II. - Fundamental properties of dielectric bodies 349 § 1 - Electrical equilibrium on poorly conducting dielectrics. 349 § 2 - Specific inductive power 352 § 3 - Electrical equilibrium on a conductive body placed in the presence of an electric current of dielectrics 352 § 4 - Properties of a capacitor with a dielectric insulating plate. Measurement of the specific inductive power 353 § 5 Causes of error. Experiments of Gaugain 363 § 6 Permanent currents in a dielectric 366 Chapter III. - Attractions of electrified bodies immersed in a medium dielectric 370 § 1 - Electrical distribution on conductive bodies immersed in water a dielectric medium 370 § 2 - Attractions between bodies immersed in a dielectric medium . 375 Chapter IV. - Pyroelectric crystals 383 § 1 - The equilibrium state of a homogeneous hemimorphic crystal 383 § 2 - Pyroelectric phenomena 387 § 3 - Gaugain's experiments 391 § 4 - Naturally and accidentally pyroelectric crystals .... 392 Chapter V. - Piezoelectric Crystals 395 § 1 - Release of electricity by the compression of substances crystallized SgS § 2 - Piezoelectricity of tourmaline 398 § 3 - Piezoelectricity of the plagiarized quartz l\oo BOOK XII. The deformations of polarized bodies. Chapter I. - The pressure inside polarized fluids 4^^ § 1 - Condition of dielectric or magnetic equilibrium of a (luid compressible 4o5 480 TABLE OF CONTENTS. Pages § 2 - Mechanical equilibrium condition of the polarized fluid 4o8 § 3 - Of the pressure inside a polarized fluid 4 '3 § 4 Change in volume of a polarized fluid 422 Chapter II. - The pressure inside V polarized solids 4^7 g 1. - Equilibrium conditions of a primitively isotropic solid, little distorted and polarized 4^7 § 2 - Pressures inside a primitively isotropic solid, little distorted and polarized 4^7 § 3 ~ Fictitious pressures inside the polarized solid 4^9 § 4 - Deformations of the polarized solid 44^ § 5 - Comparison of the results of theory with the results of experiment. 447 Chapter Kl. - Maxwell's theory 449 § 1 - History of the theory of pressures inside bodies polarized 449 § 2 - Cause of the inaccuracy of Maxwell's theory 4^7 Chapter IV^ - Electrical deformations of crystals 4^7 Table of contents 47^ FLN OF LV TABLE OF CONTENTS. 17763 Paris. - Printed by GAUTHIER-VILLARS ET FILS, quai des Grands-Augustins, 55. r, { s. \^-<. ^ f ^ > //" X -4 > ^ Mi^ \ r^ ^'■ < K \ iJ }^s 1 Q^ ^60 D77 t. 2 P&AScl Duhera, Pierre Maurice Marie Lessons on electricity and magnetism ^?^ ■k ■^ .-'^sksA -fi. 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