ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
351 
16. Again solving the equations (12) regarded as giving —^ for different values of 
OU u 
we have" 
y/RQrd </>(<) 1 
9 top (f>'(x K ) I?'(«A ®x —% 
\/ROa) 
Therefore 
m=p 
1 v 
2 ^ 
M =i 
4>'(X K ) (x k — a^faf) 
_ v /R(a; A ) iw^J M =ifr A -a M )P / («, i ) 
= 1 by (11) 
and 
p r a/ R(%) i 
a = i 1 <£'(0 %— a s 
A = 
S 
~ 2 A=l „ = 1 *x —«* 
l J - = P b [ X=p 
=i „?,Hl log d (a ‘- x ' 
= Cfi log «“* ) 
1 p =p 
— — N - 
al Slx = 1 bit 
where s may be any of the integers 1, 2, . . . , 2p-\-l. 
W riting 
^=p bctl, 
so that (16') becomes 
X p=aZ, 
p=i bu ^ 
^pf ^/B(gx) 1 1|_g 
x=i 1 Xi—a,j al s 
Weierstrass defines 
al,- s — 
al r al s — ctl s ctl r 
a r —ct, 
al r al, 
a r — a. 
al s _ al. r 
al s al r 
— _^ P J al M s 
A=1 1 0'(®a) (®A — 0(*a— a*) 
(15). 
(16') 
( 17 ) 
(16) 
(18) 
where r, .s must be different from each other, but otherwise may be any of the integers 
Of. Scott’s 1 Determinants,’ c. ix., §§ 11, 12. 
