454 
Mr. Herschel on various points of Analysis. 
where 
Y (— 0 ^ (? 
2* , ^ , 
X /n2« 2X 1 
1 ) . 7T 
in — 2 x 
1.2 .. . ( 22 :) X 1.2 .. . (zn — 22 f) . 2 ^^ * 
r ^ 2 r r 2 JP 20 r-f 1 22: 1 1 
P — ( 3 - — • ' I I 
(20) 
[+ { 5”— &c I — &c J 
If in this equation for 9 we write tt + we shall obtain the 
sum of the series 
1^” . cos 0 ’ a'*" . cos 26 ' 3*'* . cos 30 
And by addition, of 
+ 
. 2 n 
+ 
.2n 
1^” . cos 0 
+ 
3^" . cos 30 
+ 
5*” . cos 50 
+ &c ; (21 ) 
+ &c; (22) 
5. Let/ (t) = ^ and let 2 n be written for ?i and 
^ V — 1 for 6 ; and the first member of (12) becomes 
V—i . S 
. cos X 
Now the developement of — ~ — - being a t &c, that 
of/ {b) will be 
+ &c I = 
= {(1- v^=:^) + T + + &c} . + &C} 
and the coefficient of f will be found equal to 
(i_ -v/35) . ^ +' -J!p_ 
'' ' X ' I ' 1.2 ' 1.2 ... 2 ? 
Thus the application of (12) gives the following equation 
