ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
355 
so that 
gift 3 ) ' 
ffOp+r) 
P(ftp+r) 
P \a,) 
= -P(rt p+r ) 
Q(A) , 
¥(&) 
*=P 
$=1 
U s 
M's Mp +? 
P («p+r) Q(ttf) | 
. QY« p+ ,) P'(OJ 
PQp+r) 
Q'(. a p+r) V ‘ 
As a verification of these values we may deduce (14) from (14') as follows .— 
s=p >)TJ i'=p + l XTJ 
£ -Sic=dU= £ — 8u 
S=1 
bu., 
", bu-'W 
■-■ = p + l S = p 
= £ £ 
IJp+r 
Sh s 
r=p+l 7 
v, L P+ r 
bU 
r=l s=l bllp+ r B ( Ct s ) Q Zp+r) ct s a p+r j. = i Q (a p+r ) bit p ^ r 
Now the quantities St^, Sy are independent; hence we must have 
Sv. 
\TJ r=p+l 7 7 
tu 
bits r=i P / ( a dQ'( ( W) «•>■—++>• ¥>+»• 
r=p+l 7 
0= £ --V+J1 
bU 
r=l Q Zp+i ■) bu p+r 
Taking the second of these, we have 
tu 
bit 
P+r 
= -l + 
1 Z-aZp+r 
<K°W) 
and therefore 
p+r 
!• —p + 1 7 
tp + r 
r= 1 Q (Up+»') ^^p+r 
bU __--y l p+ r 
r=p+1 <f>(a p+r ) 
r=l Q (®p+r) r— 1 Q (ttp+r) 
= -1 + 1 = 0 
by the theorem already quoted in the verification of equation (11). For the first 
summation we have 
r= p +1 1 
£ 
P+r 
bU 
r=l P'K) Q X a p+r) tts — Hp+r bu p+r 
>(«■-,) 
r=p + l 7 
S ' ' P + r 
r=p + l 
. S' 
Ip+rCtl" p+r 
r = l Q ( ^p+r) tXs Cl p+r r — 1 Q (t'p-yr) (f's ^p+rf 
2 z 2 
