786] 
EQUATION. 
521 
preliminary, relating to the theory of congruences; book u. 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 Hesse’s equation, Clebsch’s equations, 
lines on a quartic surface having a nodal line, singular points of Rummer’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.” 
C. XI. 
66