MR. R. C. ROWE ON ABEL’S THEOREM. 
727 
We have then finally the following theorem. 
If i Jj(x) = 
a + bx 2 
0 (1 + nx 2 ) \/ { (1 - ® 2 ) (1 - c 2 ^' 3 )} 
dx 
then, provided that x v x. 2> x 3 are connected by the single relation 
(2— x 2 —x. 2 ~—Xo 2 + c 2 x 2 x 2 x 3 Y =4 (1 — £C x 2 )( 1 — x 2 ) (L — x s 2 ), 
we have 
’K^i) + + H x s) — \f i 
a -tan 
,-i 
■ \/ { n ( n + 1) (n + c 2 ) 
If we write sin 0 for x we have the corresponding expression 
a/ 
(n + l){n + c l \ n 
b\ _ 1 — </{n(n + !)(»+ c 2 )} sin 0 X sin 0 2 sin 0 3 
a -tan 
1 +n{l + cos 0 1 cos d 3 cos d 3 ) 
for the sum of three integrals of the form 
O 
f a + b sin 2 6 
] 0 (1 + n sin ~0 )\/(1 — c 2 sin 2 9) 
dd 
whose variables are connected by the relation 
(1 —cos ~0 [ — cos 'do — cos 2 d 3 — c 2 sin z 0 i sin 3 d., sin 2 d 3 ) 3 =4 cos ~6 l cos 2 0 2 cos 2 d 3 t. 
From the formula just proved we can deduce without difficulty the well-known 
theorems connecting the elliptic functions of each order whose variables are connected 
by the equation 
1 — cos ~dj — cos ~6 . : — cos 3 d 3 — c 2 sin ~0 1 sin 2 0 2 sin 3 d 3 + 2 cos 0 l cos 0. 2 cos d 3 = 0| 
which is only another form of the familiar relation 
cos 0 { = cos 9. 2 cos d 3 + sin 0. 2 sin 0 3 A 9 V 
* It is liere assumed that n{n+\)(ii-\-c 2 ) is positive. If this is not the case the imaginary tan - - 1 is 
replaced by a real logarithm. 
t The exceptional case 
n — 0 1 . 
h =f= 0 
in which there will be an additional term due to CM must not be forgotten. 
t We take the negative sign in the ambiguity. 
