264 Mr. Babbage's new methods oj investigating the 
c *-= 2S 7i + • a / s t§=t + 2B / s i^ + + 
The quantity may now be considered as completely 
determined, since it only depends on the co-efficients of (cos 
and the series marked by S=^, both which quantities are 
known ; the latter being given by the powers of tt and numbers 
of Bernouilli, whilst the values of the former in functions 
of n are given by Legendre, in his Exercise de Calcul Inte- 
gral, vol. iii. art. 149, 155. 
In (2,3) let x — 1, and we have 
L c = 1 
2 2 k,n i 2i (cos 6)" 
+ 
— &C. 
(2,4) 
2**(COS 20) n * 3 l ( i COS 30) n 
And if we put x = v in the other series, it becomes 
“ f ( - e2) * - <— s 2 )^— 1 , &cc ,(=«!>,. ,j_ r 7 
2 I1.2..2A + X ‘ i.z..zk—i 2 ,n* ■ ■ 1.2.3 2 k— 2, n' 1 2 k,n^ 
sin 0 sin 20 , sin 20 „ . . 
T» +T ai+, &C - ( 2 .5) 
1 2 k + I {cos 0 )’ 1 2 2 ^+ , (COS 20 )" 3 2 ^+ I (COS 3 e) ? 
If n == l this series becomes 
tan 0 tan 20 
,2^+1 
+ 
tan 30 
, 2 k+ i 
— &C. 
( 2 . 6 ) 
i 2 *+ 1 z y 
The series (2,4) may be changed into another, which con- 
tains sines both in the numerator and the denominator, for 
it is equal to 
2 2 k,n l 2k \sin0. cos0 
sin 
n l I sin 20 , 1 / sin 3 9 „ 
2 ri \ sin 20. cos 20/ * 3 14 \ sin 3©. cos 36 / 
But this becomes, since sin Q. cos 0 = i sin 2d 
2 k,n 
— 1 1 
\ sin 0 
^sin 20 
1 /sin 20 \n 
sin 30\» 
sin 60 / 
&C. 
(2.7) 
By applying the theorems [a], ( b ), (r), and ( d ) to the series 
whose sums we have now investigated, we shall arrive at the 
value of many others which contain the powers of tangents 
