CLEBSCH, kieDS11, RUDOLF FRIEDRICH AL FRED A German mathematician, born at Kiinigsberg, Prussia. He studied at K;inigs berg, where he was a pupil of Hesse. Riehelot, and F. Neumann. He held the chair of theo retical mechanics at the polytechnic school in Karlsruhe from 1858 to 1863; was Ina de profes sor of mathematics at Giessen in 1863. and at in 1868. Ills attention was drawn to algebra and geometry hy the study of Salmon's works. In 1868 he founded, with Neumann, the itathematische Annalen. His vast contributions to the theory of invariants: his use of the notion of the deficiency of a curve: his applications of the theory of elliptic and Abelian functions to geometry and to the study of rational and elliptic curves, have secured for him a pre eminent place among those who have advanced the science of geometry. His works upon the general theory of algebraic curves and surfaces began with the determination of those points upon an algebraic surface at which a straight line has four-point contact. Clebsch undertook to render the notion of 'deficiency' fruitful for geometry—a notion found first ill RiC111:1111I'S Theorie der Abelsrhen Funktionen ( 18571. By `deficiency' of a curve is meant the difference be tween the number of its double points and the maximum number possible in such a curve. (See
CUR% ES.) Clebsch and Cremona studied the representation of cubic surfaces On a plane through a one-to-one correspondence—a notion that has led to the study of higher correspond ences between surfaces by Cayley (q.v.) and Niither. Clebsch solved, by aid of the addition theorem of elliptic functions (see letiNcrioxs), the generalized form of Malfatti's Problem. lie also solved ( 0662) the so-called Tfittlian Prob lem' of differential equations, by making it de pend upon a system of simultaneous linear partial differential equations whose statement is possible without integration. Clebsch took a leading part in showing the great significance of the theory of inva via id's for the theory of hyper elliptic and functions; and to 111111 is due the transformation of the theory of binary to that of ternary forms. (See FORMS.) He died at Guttingen. November 7, 1872. Vol. vii. of the Mat hematisehe Amadeu contains an excellent article on Clebseh. in which the value of his works is estimated by Brill, Gordan, Klein, Mayer. Nuttier, and other contemporaries. llis l'orlesuagen fiber Geometric were edited by Linde mann (Leipzig, yids i.. 1875-76, vol. ii., 18911.