gave the development in series of the integrals for the particular case of linear equations. Poincaré did the same for the case when the equations are not linear, as also for partial differential equations of the first order. The developments for ordinary points were given by Cauchy and Madame Kowalevsky.
The attempt to express the integrals by developments that are always convergent and not limited to particular points in a plane necessitates the introduction of new transcendents, for the old functions permit the integration of only a small number of differential equations. Poincaré tried this plan with linear equations, which were then the best known, having been studied in the vicinity of given points by Fuchs, Thomé, Frobenius, Schwarz, Klein, and Halphen. Confining himself to those with rational algebraical coefficients, Poincaré was able to integrate them by the use of functions named by him Fuchsians.[81] He divided these equations into "families." If the integral of such an equation be subjected to a certain transformation, the result will be the integral of an equation belonging to the same family. The new transcendents have a great analogy to elliptic functions; while the region of the latter may be divided into parallelograms, each representing a group, the former may be divided into curvilinear polygons, so that the knowledge of the function inside of one polygon carries with it the knowledge of it inside the others. Thus Poincaré arrives at what he calls Fuchsian groups. He found, moreover, that Fuchsian functions can be expressed as the ratio of two transcendents (theta-fuchsians) in the same way that elliptic functions can be. If, instead of linear substitutions with real coefficients, as employed in the above groups, imaginary coefficients be used, then discontinuous groups are obtained, which he called Kleinians. The extension to non-linear equations of the method thus applied to linear equations has been begun by Fuchs and Poincaré.
