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 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.

The fertility of the conceptions of Cauchy and Riemann with regard to differential equations is attested by the researches to which they have given rise on the part of Lazarus Fuchs of Berlin (born 1835), Felix Klein of Göttingen (born 1849), Henri Poincaré of Paris (born 1854), and others. The study of linear differential equations entered a new period with the publication of Fuchs' memoirs of 1866 and 1868. Before this, linear equations with constant coefficients were almost the only ones for which general methods of integration were known. While the general theory of these equations has recently been presented in a new light by Hermite, Darboux, and Jordan, Fuchs began the study from the more general standpoint of the linear differential equations whose coefficients are not constant. He directed his attention mainly to those whose integrals are all regular.