266 Mr. Babbage's new methods of investigating the 
In the equation (1,4,) if we make ^x = tan —I x, we shall find 
. n . z n . x — z C — i 2 z-\-n — i — 2 z — n 7 
(—2) ( — 1) 2 ( — 1 ) {tan 77 + tan v } — 
a n 
= (- 2 ) (- 1 ) 2 (- 1 ) ' 
+ 
x+ x 
— 1 
71 -5 f roc 
i(cos 0)” 3 (cos 3©)" * 5(cos 56)" 
If the integrations here indicated are performed, it will be 
found that all the constants vanish, and ultimately that 
x+x~ 
X 5 +X 
—3 
If X = 1 
1 (cos 0)” 3(cos 30)” ~ 5 (cos 5 0)" 
+ 
X S -j- x 5 
&C. 
+ 
5(cos 50)” 
— &C. 
4 i(cos0)” 3 (cos 30)* 
dx 
If we multiply (3,5) by — and integrate, we have 
T log * + c = 
X — X 
t 3 
+ 
r 5 — x 5 
(3.5) 
(3.6) 
&c - (3.7) 
l (cos0)” 3*(cos 38)” 1 5*(cos 50)" 
If x = 1 C = 0, let a: = cos 6'+ V-i sin O', then we have 
w0' sin 0' sin 30' , sin 50' 
2 
4 - 
— &c. 
(3,8) 
i*(cos 0)" 3 (cos 30)” * 5 (cos 5©)" 
The equation (3,7) may be multiplied any number of times by 
— , and integrated ; and the constants thus introduced may be 
determined in the same manner as those of the equations (2,3); 
these operations will give the values of series of the following 
form : 
x+x 
, — 1 
X^ -\-X 3 
+ 
c 5 -f x 5 
j2& + I ( cos 0) H 2 2 ^^' i (cos 30) tt s 2 k +\cos$) H 
— &C. 
x — x 1 — x 1 1 * 5 — x 5 
• -7 — h — ; CxC. 
i 2 *(cos 0)” 3 2 \cos 30) 5 2 *cos 5© 
Numerous other series might be found by satisfying the equa- 
tion -tyx -|- ^ -j =J, whose sums would be given by this pro- 
cess ; and if instead of putting for x, we had substituted ax 
for x, where the function u is determined by the equation 
ofx = x , many others would be discovered. This artifice is 
