sums of several classes of infinite series. 261 
hence b = 0 , and we have 
1 _ *_±£ _ a _l±Z + fl±i! — & c . ( 1 , 6 ) 
* cos 9 cos 26 1 cos 39 v 
Continuing to integrate, it will be found that all the constants 
are zero, and we shall arrive at the following theorem; 
-1 
x-\-x x+x 
, -3 
+ — &C. 
-3 
1 (COS 0 ) 71 (cos 20/ * (COS 30) 
Let x — — x, then it becomes 
_ 1 — J+ ~ xI J_ - x * + ~* _l_ _L &c 
1 (cos 9)" (cos 29)" “ (COS 39)" • 
Putting x = 1 in both these, we have 
J.S- 1 ! L 
2 (COS 9 y (COS 29)” ‘ 
+ 
+ 
(cos 39 )” 
I 
&C. 
(1.7) 
(1.8) 
0 , 9 ) 
( 2.0 
(COS 0)° (COS 20). ”1 (COS 30). ”1” 
I propose in the next place to determine the value of the 
series 
i 2 ^ (cos 9)” 
2 Z ^(C0S 29)" 
+ 
3 2 *(C 0 S 39 )” 
— &C. 
This may be accomplished by multiplying (1 ,7) by and in- 
tegrating; this operation, being performed on it 2* times, will 
produce the series: whose sum is required ; the first integra- 
tion gives 
— A 
log X . X X 
i » ^ i (cos 9)" 
— 2 
x 1 — x 
x 1 — x 
2 (COS 29)" • 3 (COS 39)” 
If x= 1, c = 0, the second operation gives 
-2 
x % + x 
(log . c __£+£ 
1.2 ‘ ^ 1 2 (cos 9)” 
2* (COS 29)" 
— & c. 
See. 
If W = 1 
2,71 
4- &c. 
I 4 (COS 9)" 2 Z (cos 29)” 
In order to determine c 2 n put x == cos 0 + v 7 — 1 sin 9 , then 
we have 
