sums of several classes of infinite series. 
277 
t 
(sin 0)* 
— 1 I L_ . 
(sin 20)" ' (sin 30)” 
&C. 
when n is an even number, nor will the method now em- 
ployed enable us to find its value when n is odd ; the reason 
of this is that 
(sin 9 )-”= (4 
0 s 
1 . 2 . 3 . 4.5 
is developable in a series proceeding according to the even 
powers of 9 only when n is an even number, if n is odd, it 
proceeds according to the odd powers. 
The theorems contained in this second part are applicable 
to a very extensive class of series which have not, I believe, 
yet been considered. In (A) let f( 9 ) = cos 2 9 = cos (cos 9) 
and putting k — o we have 
c -~ = cos* 9 — cos 2 2 9 -f- cos 2 3 9 — &c. ( 4,1 ) 
The definite integral which is the criterion of the truth of this 
value, is 
-0 0 
fdv cos 
= A + 
—» + 
C 
9 
+ — 
~ O 
+ &C. 
1 + 2- 
7 T 
I+4 r 
7T 
1+6- 
7T 
+ 
B + — 
0 T - 0 
+ \ 
+ &C. 
I 2 - 
7T 
4 r 
1—6- 
7T 
And this is always finite unless 9 is an even submultiple of tt; 
if we make k = 1 and k = 2, we shall have the following 
theorems, which are true with the same restrictions. 
0 ±1 . sin 1 0 2 cos *0 cos *20 , cos a 20 „ 
COS 1.0--^ . — = — SiC. 
i* 1 2 2 l 2 2 l * y 
cost. S=-+fi s ^S^_l‘(ii2-L4- c 4.') = 
i 4 ‘ 2 l z 2 \ 1.2. 3.4 “ 8 / 
_ COS COS a 20 J COS 4 30 
I 4 2 + * 3 4 
&c. 
( 4 . 2 ) 
(4 >3 
If f( 0 ) = (cos n 9 ) m , since this is capable of being developed 
