24 at REPORT—1904. 
automatically into Riemann’s expression for the differential of the integral 
of the first kind, is necessarily involved in his discussion of this particular 
ease of Abel’s theorem ; it forms, indeed, the natural link between those 
results in the theta-functions already alluded to and Abel’s theorem 
itself. Thus the theory of point-groups was originally provided with a 
purely transcendental foundation—the whole superstructure being based 
upon Abel’s theorem—although, as will subsequently be seen, its true 
foundation is algebraical, and it can be rendered perfectly independent of 
transcendental considerations. 
The basis of Riemann’s proof of Abel’s theorem is his division of 
algebraic functions into classes pertaining to the same Riemann surface. 
For this enables him to take ¢ any rational function of s and z as the 
independent variable in w, an integral of the first kind on the surface 
nm 
associated with F(s,<)=0. If, then, ¢ has m infinities on the surface, 
dw . : : 2 : 
-— is anm-valued algebraic function of ¢ ; and if w®, w®,. . . ,zv™ are the 
dz 
k=m dayk) 
m-values of w for any one and the same value of Z, 3 = is a one- 
k=1 a 
valued function of ¢ whose integral is everywhere finite on the surface. 
This function, composed of the sum of m-integrals, is thus necessarily 
equal to a constant, and by proper choice of the path of integration can 
he easily shown to be zero ; with the notation previously introduced by 
Riemann, we have 
A é p(s 1%] )dz I “$(8%)dz> P(8n%m)A2m—() 
i | OF ea) | “OF Gam) 1° '* | OF Gxt). 
0s; O85 O8m 
where (s,2,) . . . (8%) are pairs of values of s, 2, at which ¢ assumes 
one and the same value.! This is Abel’s theorem for integrals of the 
first kind. Its importance for the theory of point-groups lies in its esta- 
blishment of a system of p differential equations formed by writing con- 
secutively ~, . . ¢, for in the left-hand side of equation A ; these ¢’s 
being the p linearly independent numerators in Riemann’s expressions 
for the p linearly independent integrals of the first kind. In discussing 
the integration of these equations Riemann introduces the notion of a 
system of quantities being congruent, with respect to certain moduli, to 
another system ; the p quantities (b, .. . b,), namely, are said to be 
congruent to the p quantities (a, . .. a,) with respect to 2p systems 
9 
a “=p 
of moduli, when b,=a,-+-=m,k” where r=1... p and m, ... my, are 
v=1 
integers. The notation is (b, ... b,)=(a, ... a,). 
The necessity for this notation arises—although Riemann does not 
explicitly say so—where the paths of integration in equation (A) are 
arbitrary ; the sign of equality in that equation must then be replaced by 
a sign of congruence. 
In an earlier section (§ 10) of Part I. of the ‘Theory of Abelian Func- 
tions’ Riemann had shown that a rational function can be expressed as a 
quotient of two ¢-functions provided that the number of its infinities be 
less than p+1. This result is obtained by counting constants in the 
} Riemann’s ‘ Theorie der Abel’schen Functionen,’ § 14. 2 Loe. cit. § 15. 
