» pe Pie i 
ON THE THEORY OF POINT-GROUPS. 27 
which the values of s, z are s,, 2,; in particular, if 7,.. .%, are the 
p zeros which every 9-function is shown to have, then aj? is the value 
of w, (for w=1,.. .p) at the point », (for v=1,.. .p) for which the 
values of s, z are o,, ¢,.! 
Moreover, if in the argument of the $-function the integral w/? 
or «® occurs, it is possible, by allowing the point ~%, or @, (still de- 
fined as above) to be a variable point on the surface, to consider 
the $-function as a function of z, or of Z, instead of as a function of 
the quite unspecialised point z; this is important in connection with the 
identical vanishing of the $-function. The introduction of additive 
constants ¢,... ¢, into the arguments of the $-function is another 
important feature of Riemann’s discussion of the subject ; for he shows ! 
that in $(.., u,—e, ...) it is always possible so to determine the 
= i 
lower limits of the integrals that (...¢,...)=(... Zal?. 
1 
shall hold, and it is with ¢hese lower limits that he works. The establish- 
ment of this congruence between the additive constants of each integral 
and the sum of its p values at the p zeros of the 9-function Jeads to the 
preliminary discussion in § 23 and § 24 of the conditions under which 
a 9-function vanishes identically, ie., for an arbitrary position of the 
variable point on the surface. 
»—1 
He first shows, in § 23, that if (...u,—e,...)=(... — Sa? coe 
1 
then the $-function with these arguments vanishes identically, ¢.e., for 
any arbitrary position of ¢; and, conversely, that if 3 vanishes identi- 
cally then each of its arguments w,—e¢, must be congruent to a negative 
sum of the values of p—1 integrals at certain p—1 points m . . . m4” 
Now these p—1 points may be arbitrarily chosen, and we still have 
p-l 
2 ( ...— 3a”... } identically zero; and, since $ is an even function, 
1 
p-1 
meanis leads to 9 ( ... Ba? ... ) being also identically zero ; whence 
1 
by the above converse we are led to certain p—1 points 9, . - + M-2 
p-1 a 2p—2 : 
Beem A Sat! oy Ss ( Be ers he ) sie, tothecongruence 
1 Dp 
2p—2 
( tees Ba” ... )=(0,0,...0). But this shows that the last p—1 
z 
points are dependent upon the position of the first p—1l points in 
such a manner that as the latter vary continuously we always have 
2p-2 
> da’ =0 ; and this system of differential equations is, as has been seen, 
1 
always satisfied by the 2y—2 moveable zeros of the ¢-function (the 
i ci x @ 
lower limits are another set of moveable zeros, since = = and when 
52 
€=0, 9% must = 0 but not ~®). Hence we have the important result 
that when a 8-function vanishes identically its p zeros are tied by a 
p-function. 
In § 24 a second important conclusion is derived from the fact that if 
1 Loe. cit. § 22. 
2 The precise determination of these p—1 points is as _ follows:—It 
is assumed that although & (...7,...) vanishes identically, yet that 
++ U—a+7,. ..) does not vanish identically where np is arbitrary—the 
remaining y—1 zeros of this S aren .. . mp-1. 
