252 Mr. Babbage’s new methods of investigating the 
or else they depend partly on these and partly on other 
series, containing the powers of the sines or cosines of an arc 
in arithmetical progression in their numerators, which is a 
species whose sums are easily found. For the sake of brevity, 
I shall make use of the general term of any series with the 
characteristic S prefixed to it to denote that series. Begin- 
ning then with the series SA^ we observe that when 
m is an even number, we have 
SAx''^ii"= 
(cos 20)* 
:SAX 
f i— (cos id ) 2 - 1 
.i 1 
m 
(COS 201* 
=SA r l 
SAx‘ 
i (COS 20}” 2 
+ 
222.722 — 2 
2.4 
SA-t' — - — 
i (cos 10)” + 
— &c. 
(«) 
this series will always terminate when m is an even number ; 
and if m is greater than n, the last term will have no cosines 
in its denominator: if m=n, the last term will be SAa:' - ; and if 
m is less than the last term will be SA#' — l - ; so that in 
i (COS 20)”~ m 
all cases when m is an even number, the series in question 
will depend on series of the form SAx' — 1— , or on others 
i f C0S 1 ^ 
whose sums are known. 
Let us now consider the case of m = an odd number; then 
we have m—i 
SA^ 
(sin i 0 ) m 
(cos 20)* 
SAjt' 
sin 20. | i— (cos / 0) 2 | 2 
(cos 20)* 
=SAjr'' 
(sin i 0 
m— i 
(cos 20)" 
SAx' 
sin 20 
(cos 20) 
772- 
22 — 2 
- - SAx y ■ 
sin 20 
(cos 20) 
72 — 
This series always terminates when m is an odd number ; and 
in a similar manner we shall find the two following : 
