q6o Mr. Babbage’s new methods of investigating the 
—2# 
-4* 
series -tyv ~ = Av + Av + Av + &c. multiplying by ( — 1) 
i 2 5 
and integrating, we shall get the expression 
(-*)’ (-*)* S”(-iT +T- ’= A 
(cos 0)" 
— 
V 
(COS 20 )" 
= A 
2 ' ' 3 
£ 4- A — - 4 - A - 4 - &r 
(COS 0 )" ~ 2 (cos 20 )" ~ (COS 30 )' 1 * 
(cos. 30)" 
(m) 
There occur very few cases in which it is possible to execute 
the integrations in (1,3) and (1,4); by adding the two toge- 
ther, we have 
(— »)>(— 1)*2"(— i7{V” + "+^“-”}=A^+A*‘ +r 
+ A 
x 3 +x 
(cos 30) 
• 4 - &c. 
n I 
(cos 20)" 
( 1 . 5 ) 
Here we may observe that the new series is exactly double ei- 
ther of the others ( 1 ,3) or ( 1 ,4) when x = 1 ; also, that the inte- 
gration on the left side can be executed any number of times, 
whenever $v 2x+n ypv“~ 2z ~ n is a constant quantity; the forms 
of the function ip, which fulfil this condition, have already 
been given. Let then A = 1, A = — 1, A= 1, &c. 
l+x 1 2 3 
and since >pv zz + n ■>pv'~ zz ~ n = 1, we have 
-1 , - z , -3 
x + x X + X . X 3 +X . o 
T + &C - 
&c. 
( — -2) ( 1) 2 ( ’l) - (cos 0 )” (cos z9) n * (cos 30) a 
These integrations are easily executed ; and commencing with 
72=1, we have 
• (-*?£=£ + & (-!)■= 1 +* K-iY= - £ | + 
In order to determine the constant b, put x = cos 9 + V — 1 
sin 9 ; then, since % in that case becomes we have 
1+26 V— 1 2 2 2 — - 2 + &c. = 1 
&c 
