Page:A History of Mathematics (1893).djvu/363

 If the variable be made to describe all possible paths enclosing one or more of the critical points of the equation, we have a certain substitution corresponding to each of the paths; the aggregate of all these substitutions being called a group. The forms of integrals of such equations were examined by Fuchs and by G. Frobenius by independent methods. Logarithms generally appear in the integrals of a group, and Fuchs and Frobenius investigated the conditions under which no logarithms shall appear. Through the study of groups the reducibility or irreducibility of linear differential equations has been examined by Frobenius and Leo Königsberger. The subject of linear differential equations, not all of whose integrals are regular, has been attacked by G. Frobenius of Berlin, W. Thomé of Greifswald (born 1841), and Poincaré, but the resulting theory of irregular integrals is as yet in very incomplete form.

The theory of invariants associated with linear differential equations has been developed by Halphen and by A. R. Forsyth.

The researches above referred to are closely connected with the theory of functions and of groups. Endeavours have thus been made to determine the nature of the function defined by a differential equation from the differential equation itself, and not from any analytical expression of the function, obtained first by solving the differential equation. Instead of studying the properties of the integrals of a differential equation for all the values of the variable, investigators at first contented themselves with the study of the properties in the vicinity of a given point. The nature of the integrals at singular points and at ordinary points is entirely different. Albert Briot (1817–1882) and Jean Claude Bouquet (1819–1885), both of Paris, studied the case when, near a singular point, the differential equations take the form $\scriptstyle{(x-x_0)\frac{dy}{dx}=\int(xy)}$. Fuchs