ay 6 Mr. Babbage's new method of investigating the 
+ 
. — &c. 
(COS 0)” (COS 2 0) B T (cos 30)” 
From the theorem (A) we may readily determine the value 
of the series 
l ai (COS 0)“ 2 2 *(C0S 20)” "T” 3 l4 (COS 30)" 
+ 
— &C. 
= A' 4- A' 9 2 + B' 0 4 4- C &c. 
(cos 0 )" 1 n 1 n n 
For let f(0) 
And the sum of the series required will be 
S Al -|_ A' 0’S -4 1 - + B' 0*S-4 i - + &c. 4- 0“K' . i 
1 n „ f 2A-4 • 1 n 2 
Which is precisely the same sum as we have already found 
in (2,4), except that we now find that it applies to fractional 
or surd values of n, as well as to whole numbers. 
Let us next suppose f(0) = (tan 0) z/+I = T,0 2/+1 -}- T^ 2/+3 
+ T 5 0 2/ +5 + &c. this give the series 
(tan 0 ) 
2/+ 1 
(tan 20)^+* (tan30) 2 ^+* 
T 
zk-\-i 
&C. = 
j 2 A +1 2 2k + l 3 
= T . es -ife + T 3 93s 7r^b) + &a 
The definite integral in this case being 
2 — 
9T 
r— !+* 'izi+i 
1 0 l 
p=°l 
l J 
U =.’ iJ 
= ( 7I7?^Tl A+ 7il + ^i + 75l +&c - 1 
7 F 2 dp 
Since ~~ I +v = A + Bz> ,r -|-Cz> w + &c. this is always finite 
i + v2 V 
if 9 is positive, because it is less than A + B + C -+• & c. which 
is equal to zero. 
In the former part we could only determine the value of 
the series 
