28 REPORT—1904. 
vanishes identically each of the arguments is congruent to the negative 
sum of the values of p—1 integrals at certain p—1 points ; these points, 
namely, are assumed to be p—1 of the p points in which any rational 
function takes one and the same value on the surface, and then with 
the notation explained above we have 
oe 
(p) —_— =.) v 
eine ee. =e... )=(--. aU oes 
& z , 
M4 eC) 
4 = 
Ves, Sear eran fy (ae args | 
1 
Pp (v) 
Hence for all continuous variations of Sp) &) we have du, =0,and there- 
fore the p pairs of quantities s,,~, are p of the moveable zeros of o=0, 
wu’ where the remaining p—2 are fixed. And if w?*-+-+be the values of 
2n—2 
u, at these p—2 fixed points we have ic sas Yu, ; . =(0,0,...0). Whence 
1 
2p—2 
it follows that (...¢,...)=(...—3w,...). Which results are thus 
p+i 
stated by Riemann: ‘An arbitrary system of quantities (.: 6/08 sory: 
l ee 
congruent to one system of the form {... Xa...) unless it is congruent to 
9 
p-2 
one of the form i --—Za...}), in which case it is congruent to an in- 
finite number of the first-mentioned form,’ 
It is in these results that we find the first suggestion of a point-group— 
that is, of a set of points on the Riemann surface which are chosen in 
some definite manner out of the set in which a rational function assumes 
one and the same value. Moreover, in the most general form into which 
Riemann threw these same results in his later memoir—now to be de- 
scribed—we find a conspicuous feature to be the reversible relationship which 
exists between a pair of point-groups in the two cases which he considers— 
a relationship, namely, concerning the number of points in each point- 
group which may be arbitrarily assumed. This relationship is intimately 
connected with the Riemann-Roch theorem—although Riemann himself 
was not concerned to point this out—andis a particular case of the Theorem 
of Reciprocity established by Brill and Noether. 
The first two sections of the memoir on the vanishing of the 
S-functions are occupied in putting the theorems of § 23 upon a more 
rigorous foundation by showing that it is always possible to take such 
arguments for a 9-function as to ensure that it does not vanish identically 
in which case the results of § 23 and § 24 are true. The third section 
then goes on to establish these in a still more general form, by considering 
successive pairs of 3-functions with arguments that differ from each other in 
an analogous manner to those of 9(...7,...) and &(...u,—a2+r,... ), of 
which the first vanishes identically, but not the second, and where r, itself 
is of increasing complexity. Thus a typical pair of $-functions is 
(1) SC... a PtP 4 oP. fam —yP-Y— 4-9, , , —uP-™ De, , ey 
and 
(2) Bes ae alt — PD — ye Po ey 
the first of which is assumed to vanish identically, while the second, 
whose arguments only differ by the addition of a?-™*?—w?-™, does not. 
Since (2) does not vanish identically we have, by considering it as a 
