26. The modern work, reproducing the theories of Galois, and exhibiting the theory of algebraic equations as a whole, is C. Jordan's _Traite des substitutions et des equations algebriques_ (Paris, 1870). The work is divided into four books--book i., preliminary, relating to the theory of congruences; book ii. is in two chapters, the first relating to substitutions in general, the second to substitutions defined analytically, and chiefly to linear substitutions; book iii. has four chapters, the first discussing the principles of the general theory, the other three containing applications to algebra, geometry, and the theory of transcendents; lastly, book iv., divided into seven chapters, contains a determination of the general types of equations solvable by radicals, and a complete system of classification of these types. A glance through the index will show the vast extent which the theory has assumed, and the form of general conclusions arrived at; thus, in book iii., the algebraical applications comprise Abelian equations, equations of Galois; the geometrical ones comprise Q. Hesse's equation, R.F.A. Clebsch's equations, lines on a quartic surface having a nodal line, singular points of E.E. Kummer's surface, lines on a cubic surface, problems of contact; the applications to the theory of transcendents comprise circular functions, elliptic functions (including division and the modular equation), hyperelliptic functions, solution of equations by transcendents. And on this last subject, solution of equations by transcendents, we may quote the result--"the solution of the general equation of an order superior to five cannot be made to depend upon that of the equations for the division of the circular or elliptic functions"; and again (but with a reference to a possible case of exception), "the general equation cannot be solved by aid of the equations which give the division of the hyperelliptic functions into an odd number of parts." (See also GROUPS, THEORY OF.) (A. Ca.) BIBLIOGRAPHY.--For the general theory see W.S. Burnside and A.W. Panton, _The Theory of Equations_ (4th ed., 1899-1901); the Galoisian theory is treated in G.B. Matthews, _Algebraic Equations_ (1907). See also the _Ency. d. math. Wiss._ vol. ii. FOOTNOTES: [1] The coefficients were selected so that the roots might be nearly 1, 2, 3. [2] The third edition (1826) is a reproduction of that of 1808; the first edition has the date 1798, but a large part of the contents is taken from memoirs of 1767-1768 and 1770-1771. [3] The earlier demonstrations by Euler, Lagrange, &c, relate to the case of a numerical equation with real coefficients; and they consist in showing that such equation has always a real quadratic divisor, furnishing two roots, which are either real or else conjugate imaginaries [alpha] + [beta]i (see Lagrange's _Equations numeriques_).