We have seen that among the earliest of the several kinds of "groups" are the finite discontinuous groups (groups in the theory of substitution), which since the time of Galois have become the leading concept in the theory of algebraic equations; that since 1876 Felix Klein, H. Poincaré, and others have applied the theory of finite and infinite discontinuous groups to the theory of functions and of differential equations. The finite continuous groups were first made the subject of general research in 1873 by Sophus Lie, now of Leipzig, and applied by him to the integration of ordinary linear partial differential equations.
Much interest attaches to the determination of those linear differential equations which can be integrated by simpler functions, such as algebraic, elliptic, or Abelian. This has been studied by C. Jordan, P. Appel of Paris (born 1858), and Poincaré.
The mode of integration above referred to, which makes known the properties of equations from the standpoint of the theory of functions, does not suffice in the application of differential equations to questions of mechanics. If we consider the function as defining a plane curve, then the general form of the curve does not appear from the above mode of investigation. It is, however, often desirable to construct the curves defined by differential equations. Studies having this end in view have been carried on by Briot and Bouquet, and by Poincaré.[81]
The subject of singular solutions of differential equations has been materially advanced since the time of Boole by G. Darboux and Cayley. The papers prepared by these mathematicians point out a difficulty as yet unsurmounted: whereas a singular solution, from the point of view of the integrated equation, ought to be a phenomenon of universal, or at least of general occurrence, it is, on the other hand, a very special and
