Applied to equations of the first order, the following theorem results. If an equation of the first order is reducible, only four cases are possible: 1st, the equation is algebraically integrable; 2d, it has an algebraic integrating factor; 3d, the logarithm of the integrating factor has algebraic first derivatives; 4th, a first integral is given by a system of differential equations whose general solution is of the form aui+ (a, b, c, d being arbitrary constants,) cu.+ d and which may be reduced to a Riccati equa tion.
Irreducible equations of the first order lead to known results, if we confine ourselves to the case that y shall be a uniform function of x. ' This is not the case, however, for equations of higher order. Among the equations of the second order, the simplest case is that of the equation (39) y" = Its general integral is a uniform function of x, which may be represented as a quotient of two integral transcendental functions in the form y = d2 log u is an integral trans cendental function which satisfies the equation (40) 2 + — ' where - and which may, therefore, be represented by an ordinary power-series con vergent for all values of x.
Although great progress has been made in this direction and although greater progress is to be expected as the efforts of mathematicians are being gradually rewarded, the results are meager from the point of view of the mathe matical physicist, who would like to refer to the mathematician the questions connected with the integration of a differential equation which may have appeared in some of his investiga tions. For very rarely will it happen that such an equation belongs to one of the classes with which the mathematician is prepared to deal. It remains necessary to study such equations directly by methods of successive approximation especially adapted to them, usually upon the assumption that all of the variables that enter be confined to real values. The restriction to real variables in such cases, the systematic add rigorous application of the method of successive approximations, has been productive of many valuable results in recent years, especially in the hands of Picard and Hilbert. The theory
of partial differential equations, primarily, has made rapid progress through their efforts and many mathematicians are following their ex ample. It may, however, be predicted that, even in the theory of partial differential equations, the restriction to real variables will gradually pass away. For in the case of analytic func tions, and these after all are the most import ant, the characteristic properties are veiled by such a restriction. But a necessary prerequisite for a theory of partial differential equations with complex variables is the theory of func tions of several complex arguments; this theory, however, is still in its infancy.
A. R. Forsyth, 'A Treatise on Differential Equations' (3d ed., London 1903). This contains an account of the more elementary parts of the theory. A. R. Forsyth, 'Theory of Differential Equations' (6 viols; Cambridge 1890). In this work the main stress is laid upon the functiontheoretic point of view. S. Lie, 'Vorlesungen iiber Dif ferentialgleichungen mit bekannten infinitesi malen Transformationen,' bearbeitet and he rausgegeben von G., Scheffers (Leipzig 1891). E. Picard, d'analyse' (Vols. 2 and 3; Paris 1893-95). Important principally for the modern theory of partial differential equations. P. Painleve, 'Lecons sur la theorie analytique des equations differentielles, professies a Stock holm,' Sept., Oct., Nov., 1895. Lithographed at Paris, 1897, an account of the most advanced points of view. L. Schlesinger, 'Handbuch der Theorie der linearen Differentialgleichungen' (Leipzig 1895-98). For further bibliographical indications and for a brief account of the whole subject from the point of view of the expert mathematician, see the various articles an dif ferential equations in der mathe matischen Wissenschaften mit Einschluss ihrer Anwendungen,' which is now being published in Leipzig. A French edition, much more ex tensive than the German edition, of this monu mental work is also being published.