ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
861 
Inverting it in order to obtain s in terms of u’s we find that to the fifth order 
. A* 3 . Lr p u.J> 
d" o u »t ~r 2 m a,., m o 
6 ,~i 6 
,5AJ + ZB m „ , A* SI ^ a, (Mt (3a r>w +2A,) , , , ^ «/=<> , 
+ 7~Z U,n ~—U m S»a r,»{U r + %m — - U r -fi 3 ‘A r ni U/ S 1 ' S g r U^ (29) 
J-0 O r~l r=l ro ■ s=l 
where S™ implies summation for all values of r from 1 to p except m, and S>- for all 
r=l 
values of s from 1 to p except r. Substituting this in (27) we obtain 
3=1 
3 9 A 
m=p 1 
■U= S — 
m=l 1 V K»)L 3 15 
m 5 , U? . /2B m A,„ 2 
o + i d~ u r,i a,. Q fi- ( -xw -f- 
' 1 ' ,.= 1 o \ do 
u. 
3 
2 A r =p 
+ ~ a, >OT ^+-r- 2»a v «(3a,, % +2A,K 
O r = 1 O 10 ,. = i 
qi z r=p s=p r=p s=p 
+ m x' o, XT' q i w m ^ q q 
XT 2» a,. jlB M r - S'- a,,. AL+qr 2« 2«a^a S) ^X 
^ 7’= I S—l v T= 1 5=1 
• ( 30 ) 
correct to the seventh order. In the last term inside the bracket r and s may take 
the same value ; the double summation is in fact 
r = p 
2" 1 a r ,m u r > 
r— 1 
Again 
alJ[u)=j <f>(a m ) 
I'm 
—— $m'( 1 a 2i mP\) ( 1 ” 3>2i * 
(the term involving s m z not occurring in the brackets) 
(1 -a P , w v) 
= .S', 
r, i=p 
I 2 m a^ }li S r -j - S &r, m&t, m^r $t~ 
r, t=p 
where S™ implies summation for all values 1, 2, . . . , p of r and t except m, and 
r,t=1 
and t must not have the same values. Extracting the square root we find 
al m (u) —.s'. 
1 — i 2™ a,. ,„v — } S™ a,. t JV+i 2»‘ £ 
r= 1 r=l r, £= 1 
• ( 31 )- 
Let <r„ t refer to al m (y), S„, to al m (u-\-v), so that to the first order 
MDCCCLXXXIII. 3 A 
