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
defined by the position and nature of its singularities, and who has applied this conception to that linear differential equation of the second order, which is satisfied by the hypergeometric series. This equation was studied also by Gauss
and Kummer. Its general theory, when no restriction is
imposed upon the value of the variable, has been considered by J. Tannery, of Paris, who employed Fuchs' method of
linear differential equations and found all of Kummer's twenty-four integrals of this equation. This study has been continued by Édouard Goursat of Paris.
A standard text-book on Differential Equations, including original matter on integrating factors, singular solutions, and especially on symbolical methods, was prepared in 1859 by George Boole (1815–1864), at one time professor in Queen's
University, Cork, Ireland. He was a native of Lincoln, and a self-educated mathematician of great power. His treatise on Finite Differences (1860) and his Laws of Thought (1854) are
works of high merit.