The condition in order that an equation of a given prime order n may be solvable by radicals was in this way obtained--in the first instance in the form (scarcely intelligible without further explanation) that every function of the roots x1, x2 ... x_n, invariable by the substitutions x_(ak + b) for x_k, must be rationally known; and then in the equivalent form that the resolvent equation of the order 1.2 ... (n - 2) must have a rational root. In particular, the condition in order that a quintic equation may be solvable is that Lagrange's resolvent of the order 6 may have a rational factor, a result obtained from a direct investigation in a valuable memoir by E. Luther, _Crelle_, t. xxxiv. (1847). Among other results demonstrated or announced by Galois may be mentioned those relating to the modular equations in the theory of elliptic functions; for the transformations of the orders 5, 7, 11, the modular equations of the orders 6, 8, 12 are depressible to the orders 5, 7, 11 respectively; but for the transformation, n a prime number greater than 11, the depression is impossible. The general theory of Galois in regard to the solution of equations was completed, and some of the demonstrations supplied by E. Betti (1852). See also J.A. Serret's _Cours d'algebre superieure_, 2nd ed. (1854); 4th ed. (1877-1878). 25. Returning to quintic equations, George Birch Jerrard (1835) established the theorem that the general quintic equation is by the extraction of only square and cubic roots reducible to the form x^5 + ax + b = 0, or what is the same thing, to x^5 + x + b = 0. The actual reduction by means of Tschirnhausen's theorem was effected by Charles Hermite in connexion with his elliptic-function solution of the quintic equation (1858) in a very elegant manner. It was shown by Sir James Cockle and Robert Harley (1858-1859) in connexion with the Jerrardian form, and by Arthur Cayley (1861), that Lagrange's resolvent equation of the sixth order can be replaced by a more simple sextic equation occupying a like place in the theory. The theory of the modular equations, more particularly for the case n = 5, has been studied by C. Hermite, L. Kronecker and F. Brioschi. In the case n = 5, the modular equation of the order 6 depends, as already mentioned, on an equation of the order 5; and conversely the general quintic equation may be made to depend upon this modular equation of the order 6; that is, assuming the solution of this modular equation, we can solve (not by radicals) the general quintic equation; this is Hermite's solution of the general quintic equation by elliptic functions (1858); it is analogous to the before-mentioned trigonometrical solution of the cubic equation. The theory is reproduced and developed in Brioschi's memoir, "Uber die Auflosung der Gleichungen vom funften Grade," _Math. Annalen_, t. xiii. (1877-1878).