ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
349 
is expanded in des- 
P(.c) . . . V(x) 
since • v - is an integral function of x, and when ;- ' 
x-a p ° (x—a^y/Rix) 
cending powers of x the highest index of a? is —§. We have shown that a 1} a 2 , ... ,a K 
may be regarded as triple roots of the equation for the roots, and thus we may take as 
the constant 
P (x)dx 
K=p ra 
3 t 
A 
Hence 
A = p 
t 
A 
life 
T(x)dx 
a^s/T^x) 
iT— — 
= iJ (x—a pL ) x /R(x) 
^ fh P (x)dx f-P\ P (x)dx 
r J„. (x—aj\/R(x)~ r ] a . (x— 
or 
P (x)dx 
(x—a^x/Rix) 
u p +v p +iv=0 
(x—a^x/ E(» 
~0 
(7). 
Now, by Weierstrass’s theory, given values of u l} u z , ... , u p imply unique values 
of x x , x 2 , . . . , x p which are, in fact, the roots of an equation of the p th degree whose 
coefficients are single-valued functions of u v u 2 , . . . , u p . Every symmetrical function 
of x 1 , ... , x p can therefore be expressed as a function u v u 2 , . . . , u p , but in 
particular 
(a t —x 1 )(a ( —x 2 ) . . . (a t —x p ) 
(t being any of the integers 1, 2, ... , 2p+l) is the perfect square of such a function. 
Write 
<f>(x) =(x—x 1 )(x — x. 2 ) . . . (x—x p ) .(8) 
—QK) =1 (? — i, 2 ,..., P ) 1 
%u)=U^l,2,...,/)+l)J 
then Weierstr ass defines 
l } al r 2 =cf)(a r ) . 
(9) 
( 10 ) 
for all values of r included in 1, 2,, 2p-f-l. It is easy to verify that aq, x_, 
are the roots of 
r =p r l r al? 
r=i \{a r —x)R\a,) 
= 1 
x n 
( 11 ) 
for there are obviously p roots, and in order that x l may be one of these we must 
have 
r ^ (ctr-x 2 )(a r -x.^ . . . aip) _ 
r=i (a r —a^)(a r —a 2 ) . . . (cc r — a p ) 
(A). 
By a known theorem of Abel’s we have 
2 ( |=0 or 1. 
dz 
