724 
MR. R. C. ROWE ON ABEL’S THEOREM. 
so that 
f( x > y )=~ y ; y )= 2 ; M x )= 1 • 
Also let 
0{y)=y+x—a* 
Eliminating y we get 
2a; 2 —2ax+a 3 —1 = 0 
as the equation of the limits. 
If x x , x 2 are the roots of this equation we easily find 
xy-\-x, 2 ~=l. 
The theorem then gives 
r 
^ 7(1 lo ° ( y-a + x) + C 
y — a (?/ —a) 2 
.1 
— 2C4 %-[ log x-\- 
x y\ 
2x 2 
But 5f-=0, wherefore the right-hand side reduces to a constant, and we have the 
result that 
is constant if 
and so 
f*» dx 7 dx 
W(W) + W(i-* : 
Xl *+x*= 1, 
1 dx 
=1 
and this is, of course, the well-known theorem that # + if sin 2 d-f sin 3 <£=l, 
(the angles being restricted to the first quadrant). 
II. The elliptic functions. 
As a second example take 
X= 
a + bx 2 
(1 + nx 2 ) 7 (1 — x*) (1 — c 2 x 2 ) 
and let 
x(y)=y°-( l ~ x *)( t-cV) 
so that 
IX./,-1 2(a + W) rfe 
1 
4 
§ 
tc 
7 
A 
and 
/i(«, 2/) = 2(a-f6x 3 ) 
f 2 (x)=l+nx\ 
* To choose the more general form y + bx—a leads by similar steps to a less interesting result. 
