32 REPORT—1883. 
magnitude # + iy, and it is this which would have to be gone into for 
any explanation of his method of dealing with the problem. 
Riemann, starting with the integrals of the most general form, and 
considering the inverse functions corresponding to these integrals—that 
is, the Abelian functions of p variables—defines a theta-function of p 
variables, or p-tuple theta-function, as the sum of a p-tuply infinite 
series of exponentials, the general term of course depending on the p 
variables ; and he shows that the Abelian functions are algebraically con- 
nected with theta-functions of the proper arguments. The theory is pre- 
sented in the broadest form ; in particular as regards the theta-functions, 
the 4” functions are not even referred to, and there is no development as 
to the form of the algebraic relations between the two sets of functions. 
In the Theory of Equations, the beginning of the century may be re- 
garded asan epoch. Immediately preceding it, we have Lagrange’s ‘ Traité 
des Equations Numériques’ (Ist ed. 1798), the notes to which exhibit the 
then position of the theory. Immediately following it, the great work by 
Gauss, the ‘ Disquisitiones Arithmetice’ (1801), in which he establishes 
the theory for the case of a prime exponent x, of the binomial equation 
a” —1=0: throwing out the factor « —1, the equation becomes an 
equation of the order n — 1, and this is decomposed into equations 
the orders of which are the prime factors of »—1. In particular, 
Gauss was thereby led to the remarkable geometrical result that 
it was possible to construct geometrically—that is, with only the ruler 
and compass—the regular polygons of 17 sides and 257 sides respectively. 
We have then (1826-1829) Abel, who, besides his demonstration of the 
impossibility of the solution of a quintic equation by radicals, and his very 
important researches on the general question of the algebraic solution of 
equations, established the theory of the class of equations since cailed 
Abelian equations. He applied his methods to the problem of the divi- 
sion of the elliptic functions, to (what is a distinct question) the division 
of the complete functions, and to the very interesting special case of the 
leminiscate. But the theory of algebraic solutions in its most complete 
form was established by Galois (born 1811, killed in a duel 1832), who 
for this purpose introduced the notion of a group of substitutions; and 
to him also are due some most valuable results in relation to another set 
of equations presenting themselves in the theory of elliptic functions— 
viz. the modular equations. In 1835 we have Jerrard’s transformation 
of the general} quintic equation. In 1870 an elaborate work, Jordan’s 
‘Traité des Substitutions et des Equations algébriques:’ a mere inspec- 
tion of the table of contents of this would serve to illustrate my proposi- 
tion as to the great extension of this branch of mathematics. 
The Theory of Numbers was, at the beginning of the century, represented 
by Legendre’s ‘ Théorie des Nombres’ (1st ed. 1798), shortly followed by 
Gauss’s ‘Disquisitiones Arithmetice’ (1801). This work by Gauss is, 
