728 
MR. R. C. ROWE OR ABEL’S THEOREM. 
For the first kind. 
Here we put 
a= 1, b — 0, n = 0. 
This does not fall under the exceptional case; and our formula gives 
m)+F(0 2 )+F(0 3 ) = O. 
For the second kind. 
Here we put 
a=\, b=—c z , n— 0. 
This gives rise to the exceptional case. 
The right-hand side of the formula vanishes. It remains to find the value of 
1 _w2,v>2 
— Ci2——log Or, 
X J 
~ 1 — dot? , 1 + px + qx 3 — t/, 
=—Cl- -iog- 
Vi 
. +px + qx 1 —y l 
where y l = (1 — x 3 ) (1 — c¥) 
= 2Cj(l— c~x z 
X 
which, clearly, 
1 
Ih 
1 +px + qx 2 J (1 + px + qx 3 ) 3 
fl“-+.. 
! 9 X 
(c 2 .r 4 — . 
I i 
,.)(i 
_Sp \ 
9 X ‘ ‘ 7 , 
1 qF 
+ 3 
q 3 X 6 
~r 
= 2c z ( 
r\ 0 
Zpc* 
r-cr 
= —c 2 sin 0 L sin 6, sin 0 5 . 
Therefore E(0 1 ) + E(0o) + E(# 3 ) = — c s sin 6 l sin 0, 2 sin d 3 . 
For the third kind. 
We have to write a= 1, 6=0, and get 
n(», 0 1 )+n(n, 0 2 )+ll(n, 0 Z ) 
=-v; 
tan" 
n(n+ l)(n + c 3 ) sin 0 X sin 0» sin 0 3 
(n-rl)(n-\-c 3 ) “““ 1-Hfil — cos 0 1 cos 0. 2 cos 0 3 ) 
(or the corresponding logarithmic expression if n(n-\- l)(w-f-c 2 ) is negative).'"' 
* Catlet, ‘Elliptic Functions,’ art. 132. 
