The author studies systems of $n$ differential equations of second order which admit a Lie group of transformations. He remarks that a particular case of this problem is the problem of finding all the Riemannian spaces, the geodesic lines of which admit such a group; this problem has already been solved by the reviewer [Mem. Accad. Sci. Torino (2) 43 (1903); Atti Accad. Naz. Lincei. Rend. 14 (1905)].
   The first chapter is concerned with two kinds of systems of differential equations. Systems I define the second derivatives of unknown functions $x_i\ (i=1,2,\cdots,n)$ with respect to a parameter $t$ as functions of the $x_j$ and their first derivatives ($t$ not appearing explicitly in the equations). Systems II give $d^2x_i/dx_n{}^2\ (i=1,2,\cdots,n-1)$ as functions of the $x_j$ and of the $dx_j/dx_n$. Given two systems I, the author derives conditions under which it is possible to transform one into the other by means of a change of the independent variable $t$, and similarly conditions that a system I be equivalent to a system II. He finds that a special rôle is played by the systems III of $n$ equations: $$\frac{d^2x_i}{dt^2}=\sum_{r,s}f_{rs}^i\frac{dx_r}{dt}\frac{dx_s}{dt}+\frac{dx_i}{dt}R,$$ where the $f_{rs}^i$ are functions of the $x_j$ only and $R$ is independent of $i$ and a function of the $x_j$ and $dx_j/dt$. For such systems the author generalizes Riemann's symbols of the second kind and Ricci's calculus in Riemannian spaces.
   Next conditions are stated under which a given system III admits an infinitesimal transformation $$X=\sum_i\xi^i\frac\partial{\partial x_i},\quad\xi^i\text{functions of the}\,x_j,$$ and the corresponding partial equations for the $\xi$ are found. Using the conditions of integrability for these equations, the author shows that the second derivatives of the $\xi$ are linear functions of the $\xi$ and their first partial derivatives with respect to the $x_j$. There exist special systems which, by a suitable choice of the $x_j$ and of the independent variable, can be transformed into the system $x_i{}''=0$. This case excepted, the author finds the maximum number $M$ of parameters on which a group transforming a system III into itself may depend. For example, if $n\geq 3$, then $M=(n-1)(n-2)+3$. In studying the infinitesimal transformations of a group $\Gamma$ which transforms a system III into itself, he takes into account their order $r$; if Taylor's development of the $\xi$ begins with terms of degree $r$, the transformation $X$ is of order $r$. From the preceding theorems it follows that, in the most important cases, $r$ is equal either to zero or to 1. The following investigation concerns the set of transformations of $\Gamma$ of order zero, and the set $\gamma$ of transformations of order 1 (which transforms the origin into itself), and the relations among such transformations. Moreover, a study is made of the group $\overline\gamma$ generated by the infinitesimal transformations of $\gamma$, where all the terms which are not of first order in the development of the $\xi$ are omitted. Owing to the long calculations a detailed description of them is impossible. Next the author studies systems having a group $\Gamma$ depending on $M$ or $M-1$ parameters; here the properties of the corresponding group $\overline\gamma$ are important. Despite their length the calculations in the paper are not complete. Finally the author determines all the systems of three equations which admit a group. Many results are summarized in synoptic tables.