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ARTICLE 13. THEORY OF FUNCTIONS. 36 memoir (1845) on doubly periodic functions extended Abel’s investigations on doubly infinite products. Euler had given singly infinite products for sin x, cos x, and Abel had generalized these, obtaining for the elementary doubly periodic functions expressions for snx, cnx, dnx. Starting from these expressions of Abel’s Cayley laid a complete foundation for his theory of elliptic functions. Eisenstein (1847) followed, giving a discussion from the standpoint of pure analysis, of a general doubly infinite product, and his labors, as supplemented by Weierstrass, are classic.

Monge (1809) treated ordinary and partial differential equations of the first and second order, uniting the theory to geometry, and introducing the notion of the “characteristic,” the curve represented by F (z) = 0, which has recently been investigated by Darboux, Levy, and Lie. , Geschichte der Mathematik, Vol. III, p. , Handbuch der Theorie der linearen Differentialgleichungen, Vol. I, 1895, an excellent historical view; review by Mathews in Nature, Vol. LII, p. , Some of the Developments in the Theory of Ordinary Differential Equations, 1878-1893, Bulletin New York Mathematical Society, Vol.

To Jacobi is also due the conformal representation of the ellipsoid on a plane, a treatment completed by Schering (1858). The number of examples of conformal representation of surfaces on planes or on spheres has been increased by Schwarz (1869) and Amstein (1872). In 1844 Hesse, whose contributions to geometry in general are both numerous and valuable, gave the complete theory of inflections of a curve, and introduced the so-called Hessian curve as the first instance of a covariant of a ternary form.