sums of several classes of infinite series. 
253 
• (cos id ' j I— (sin z‘0) a } 2 
C A ~i l c0s q A ~i± i = 
(sin id)” ^ (sin z0)" 
=SAo: i T-^ 
nt 
(sin 20)" 
SA-r* 
.11 — 2 
m . m—2 
’ 2.4 
SAj;*- 
7 2 (sin id) 
when ra is an even number, and 
cosz'0|i — (sin 20) 2 | 2 = 
i (sin id) 
m— x 
(0 
SA^i^ = SAx i 
(sin 20)" 
1 1 
cos id m—i v k cos 20 
(sin id )" 
-SAr 
(sin id)” 
W2— 1 . m — 3 
(sin 20) 
n — 2 
2 . 4 SAi-^&c. (J) 
T i (sin 20) ^ 
when m is an odd number. 
Let us now propose to investigate the sum of the series 
Aar 
+ 
A* 2 
Z 
Ax* 
+ 7 
(sin 0)" 1 (sin z0)" 1 (sin 30)" 
Assume \|at = Ax + Ax 2 -{- Ajt 3 + &c. 
I 2 3 
Put v* z for x ; then it becomes 
= Av 2 * -j- Av Az -j- Av ** -{- &c. 
1 2 3 
Integrate both sides, observing that £?/'*== ^— then we have 
7)431 716% 
2 V = AK + A^ + A K + fc 
I 2 3 
Integrate again, and after the n tb integration we shall have 
ii' lx 7.+* n 6;c 
= ^(^ZT> + A (^— 77 * + ^ (V=T7 + &c - 
Nowletz> = cos 9 ± V — 1 sin then our equation becomes 
(z\/ — i ) n % n ^ tz = A 
(sin 0)" 
(sin 20)' 
ytX 3 m 
i" "i" A (sin 30 )" “f” ^ C - 
Put z + — for z, and we have 
• 2 ’ 
(, v - xy ^ = a^ + a 7 ^+a^ + &c. 
(sin 20)" 
L (sin 30)" 
