730 
MR. R. C. ROWE ON ABEL’S THEOREM. 
As a particular case of this result what is often called Abel’s Theorem may be 
adduced. 
Let 
x M 
(x-a)</ 0x7)027) 
We have to write in the previous work 
for 
<K X ) 
/7) 
x — ci 
for 
0(7 
0i (702(7 
for 
m 
n 
1 
2 
The right-hand side becomes 
0 
77) 
Cl> CC 
1 v 
70x7)77) o g 
CO 
70x7)027) 
f he two values ot co are + 1, -1. 
Therefore the above 
= 0 
77) 
x — a y/0x7)027) 
i 7 A> y/ 0i 7) 027) 
10 °’- . 
This assumes a more symmetrical shape if, with Abel, we write, not y/<£1(777) = 
Ao 
but A/5 so that y/0 x 
o i a*) x 2 
With this alteration we get 
7)0 3 7) = 
A0i(7 
x., ‘ 
f /7)<& Q 
77) ~ 
1 
7“«)70i7)027) _ 
x — ci 
7 017) 7 7) 
log 
7 y/0 x 7) X. ,y/037) ™ 
Xx \t 0i7) "t X, y/ 0.> (f) 
/(«) 
y/ 0x(a)0 g (a) 
log 
X] («) \/t 0i(a) X.) f ^0 \d (p 2 (ccj 
\(a) y/ 0 1 (a) + X 2 (a) y/ 0 2 (a) 
•a 
/7) 
'(x—a)y/ 0x7) </>o(«) 
log 
Xi(U.') \/ 0x7) \ 2 (x) y/ 
A7) y/0x7) T X,(®) y/(jL7) 
which is the well-known theorem referred to. 
We see it to be only a particular case of a particular case of the theorem called in 
this paper Abel’s Theorem, 
