278 Mr. Babbage's new method of investigating the 
in a series proceeding according to the even powers of 0, we 
shall have if ^ = 0 
/cos” — I t\ m 
— -- — - = (cos n G ) m — (cos n 20) w -|- (cos “ofiY — &c. (4,4) 
and the definite integral is finite in this case, whenever (5 and 
7T are incommensurable : we may therefore in the same cir- 
cumstance have the value of the series 
(cos (cos ”20)” 
+ 
(cos ”30)” 
&C. 
The theorems marked (A) and (B) in this paper correspond 
with that marked (12) in Mr. Herschel’s memoir “ On various 
points of Analysis/’ printed in the Philosophical Transactions 
for 1814; with the first of these it coincides when n is an even 
number, and with the second when it is an odd one : the 
theorem alluded to is 
-0»f(.-*®) 
+ (- 
|=-L(2). «„ 0 "+*L( 2 ). «v_/- 2 + See. 
(2 
1 — I )tt 
1.2 
ZX 
B 
zn— 1 
where °L = 1 — 1 -f- 1 — &c. = \ 2 *L(2) = 
Now this latter expression is the value of those series which 
I have expressed by S^-. Both methods give the same result, 
and as that result is very frequently erroneous, I shall con- 
firm the truth of the explanation I have offered, by shewing 
in a particular case, that if the sum of that part of the series 
which had been neglected as being equal to zero, is found 
and added to the other part, the result will no longer be 
erroneous : the example 1 shall examine is the series 
» 1 . 1 1 4. & c 
1+6* 1 + 2*6* 1 + 3*6* 1+4*0*^ 
In Mr. Herschel’s theorem, making f(e') = ctf + « 2 * a -f- 
&c. = 
1 + ®t \/ — 1 
1 +0*/* 
we have <* o = 1, and the equation becomes 
