256 Mr. Babbage’s new methods of investigating the 
By continuing to multiply (3) by ^ an ^ integrating, it is easy 
to perceive that we should arrive at the two following theo- 
rems* 
, 2k 
S zk — Z 
( lQ g-0^ I ( lQ g x t . r I &c 4- c = 
1.2..2k.Q ^ I.2...2& — 2 ’ * , 
I Zk — I 
— * 
x-\-x 
— 2, 
x z +x 
z 2k -'sin 2 6 + 3^ _, sin 3& 
2 £— - I - 
i sin 
( 8 ) 
sanj^L + te<H c +&c. +^c = 
I.2..2&+I.0 ^ I.2..2&— I 1 2^—1 
- X 
a: — a? 
i 2 ^sin 
x z —x 
2 Z ^ s 'in 20 
+ _i!=£__&c. 
3 2 ^sin 36 
the constants c, c, c, &c. may easily be determined from each 
1 3 5 
other the value of c has been already found, that of c and c 
’ 1 3 5 
are as follows : 
r — i-S — 4 -— S — + 
t ; — e o z - 4 t 5 ^ “ 360 
c =^ S f + ^s# + 2 £ 0 s^-H’ 
5 
___ QS 3i 
360 0 i 1 " 1 “ p 5040 
a variety of series are deducible from those of (8) ; I shall 
only mention two of them ; 
cot 9 cot 20 . cot 30 
zk-—l „ 2 & 1 „2A 1 
and 
sin 0' 
sin 0 
2— ' 3 
1 sin 20' 
lk ' sin 20 
+ ■ . ”i£_&c 
3 i4 sin 30 
Returning to the formulae (i) and (2) by addition and 
subtraction, we shall have when n is even 
r zz + n —2 z—nl 
(*✓— +(-# j — 
A »+»* 1 a 
s (sin 0) n ” 2 
x z +x 
(sin 20)” 
+ A 
x 3 +~ 3 
(sin 30) 
- + &C. 
