ON THE THEORY OF POINT-GROUPS. 25 
1) \ 
quotient f the reasoning is somewhat obscure, and the number does 
(2) ? 
not tally with the number of arbitrary constants previously found by the 
author ; on the contrary, it agrees with Roch’s more accurate formula, 
showing that, in certain cases at least, Riemann knew that his own 
needed modification. But, however this may be, the result itself is true, 
and its use in the integration of the p equations formed from equation 
(A) is of importance. For if ¢ is expressible as the quotient 7, (on if 
ie8) 
m<p +1, as the quotient a) then (s;%,) . . + (8n%m) are the common roots 
of F(s, 2)=0 and of C=X (or of F(s, 2)=Oandé=".). That is to say, 
(8:21) « - + (8n%m) may be regarded as the common roots of F=0 and yx=¢y 
(or of F=0 and #°—%»®=0), but they are roots which vary with ¢, not 
those which make y=Y=0 (or o?=9®=0) independently of ¢. When 
m <p+l, they are therefore m of the 2p—2 moveable zeros of a p-functron, 
since ¢°—p” obviously fulfils the necessary conditions which make it 
@-function.! It is, of course, also possible, though not necessary, that 
even when m>p-+l, ¢ should be expressible as the quotient of two 
@-functions, and in such cases, once more, (8,2) - + + (8nZm) are m of the 
2p—2 moveable zeros of a ¢-function. 
Now Riemann shows that the p differential equations formed from 
equations (A) can be completely integrated, under certain conditions : 
first, when m<p+l1 and ¢ is perfectly general ; next, when m=p, in 
which case { is necessarily expressible as the quotient of two ¢-functions ; 
and, lastly, when m=2p—2, provided that ¢ 2s expressible as the quotient 
of two ¢-functions. For, in the first case, if m=p-+1, ’ has p+1—p+1, 
i.e., two independent constants ; it therefore depends upon one arbitrary 
parameter after its +1 infinities have been chosen, and therefore s, 2, 
which are (p+1)-valued functions of ¢, also depend only upon one 
parameter, when | has been chosen so that it becomes infinite (ize. e—0) 
at the p+1 lower limits of the integrals; of the p+1 upper limits 
(8,21) - - - (Sy41%p41) Of the p+1 integrals connected by the p differential 
equations it is thus seen that only one can remain arbitrary under these 
conditions, and the system can therefore be completely integrated. If 
m <p+l1 the reasoning only needs to be modified by considering that 
certain of the upper and lower limits coincide, whereby such integrals 
drop out from the equations. The solution of the p differential equations 
may then be written 
#=p+1 p+1 pti 
BS cathy Da tose Dt wi? ) ==) (ep side) 
#=1 u 1 
where c, . . . ¢, are constants which depend upon the choice of the lower 
limits of the integrals.” 
In the second case, when m=p, ¢ is necessarily of the form a and 
now, by the Riemann-Roch formula, it has p—p+1+1, 2<., ee inde- 
1 Loe, cit. § 16. 2 Loe. cit. § 14. 
