Page:A History of Mathematics (1893).djvu/361

 a system of ordinary differential equations (in analytical mechanics) and a partial differential equation, Jacobi drew the conclusion that, of the series of systems whose successive integration Pfaff's method demanded, all but the first system were entirely superfluous. Clebsch considered Pfaff's problem from a new point of view, and reduced it to systems of simultaneous linear partial differential equations, which can be established independently of each other without any integration. Jacobi materially advanced the theory of differential equations of the first order. The problem to determine unknown functions in such a way that an integral containing these functions and their differential coefficients, in a prescribed manner, shall reach a maximum or minimum value, demands, in the first place, the vanishing of the first variation of the integral. This condition leads to differential equations, the integration of which determines the functions. To ascertain whether the value is a maximum or a minimum, the second variation must be examined. This leads to new and difficult differential equations, the integration of which, for the simpler cases, was ingeniously deduced by Jacobi from the integration of the differential equations of the first variation. Jacobi's solution was perfected by Hesse, while Clebsch extended to the general case Jacobi's results on the second variation. Cauchy gave a method of solving partial differential equations of the first order having any number of variables, which was corrected and extended by Serret, J. Bertrand, O. Bonnet in France, and Imschenetzky in Russia. Fundamental is the proposition of Cauchy that every ordinary differential equation admits in the vicinity of any non-singular point of an integral, which is synectic within a certain circle of convergence, and is developable by Taylor's theorem. Allied to the point of view indicated by this theorem is that of Riemann, who regards a function of a single variable as