ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
353 
r=p + l 
U x — 2 
V'P + r 
Up — 
r= 1 ttp+r 
r = p + l n, 
a p+r 
r= 1 « 3 — «p+r > 
r=p + l 
u = 2 
Up+r 
r= 1 ^p ^p+i' 
These new u’s are not fully determined : as tire remaining equation necessary to 
determine them assume 
f{ U P + 1’ • * • 5 M 2p + l) = °- 
When substitution is made in U for u v . . . , u p , U will be a function of u p+1 , 
and we shall have 
, u 2p+l ; 
1 tu . 
•W bu p+r ° Up+r S Z bu 
s= p ? -=p+i -y 
= 2 2 
* — Su s 
Su„ 
i r=i a s -cc p+r bu s 
and from the (p-j- l) th equation giving the new ids 
r=p+1 ¥ 
2 -c~Bu +r =0. 
?•=! 
Then by the principle of indeterminate multipliers 
hU s =i> 1 -tU t/ 
2 -— =X- y 
s— l Ct's Mp+r Xu s bu p _i_ r 
for all the p-f-1 values of r. Multiply these p+1 equations by u p+1 , u p+2 , . . . respec¬ 
tively and add; then 
¥ 
r= p +1 tU s zf> tU J = L +1 
2 u p+r - — 2 ,u s . — X 2 u 
Olip-^r s=i OlC s ?* = 1 
p+r 
Let the part of U which is of the order m in the u s ’s (s<p) be denoted by JJ m ; then 
when expressed in terms of the u p+/ ’s it still remains the term of order m, so that 
r=p +1 VTT s—p \TJ 
. U Li m TT U KJ n 
2 u p+r — —m\J m —tu s T7~ 
r— 1 °Up+r s=l OU s 
and summing up for the terms of all orders 
r=p + l iU *=? 6U 
2 U + r T- =2 U s . 
r= 1 OU p+r s ~i 6U S 
M DCCCLX XXITI. 
2 z 
