EQUATION. 
520 
[786 
possible to reduce the original group of the equation so as to make it ultimately 
consist of a single permutation. 
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 
x 1} x 2 ,...,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, 
Grelle, 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 Betti (1852). See also Serret’s Cours 
d’Algèbre supérieure, 2nd ed. 1854; 4th ed. 1877—78. 
33. Returning to quintic equations, Jerrard (1835) established the theorem that 
the general quintic equation is, by the extraction of only square and cubic roots, 
reducible to the form ¿c 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 Hermite 
in connexion with his elliptic-function solution of the quintic equation (1858) in a 
very elegant manner. It was shown by Cockle and Harley (1858—59) in connexion 
with the Jerrardian form, and by 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 Hermite, Kronecker, and 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 repro 
duced and developed in Brioschi’s memoir, “ Ueber die Auflösung der Gleichungen vom 
fünften Grade,” Math. Annalen, t. xiu. (1877—78). 
34. The great modern work, reproducing the theories of Galois, and exhibiting 
the theory of algebraic equations as a whole, is Jordan’s Traité des Substitutions et 
des Équations Algébriques, Paris, 1870. The work is divided into four books—book I.,