2 7° Mr. Babbage's wezt; methods of investigating the 
repeat the same process we have already explained, we shall 
arrive at the following theorem : 
f(g) f( 2 X) , f( 3 *) 
j2fc+I 2 2*+I ' 3 2 ^+* 42*+* ' ' 
= A *S-±| + + &C. + i K (B) 
Let f(0) be any function of 0 developeable in the form 
f(0) = A + B cos 5 + C cos 20 + D cos 3 9 & c. 
a very similar process to that which has been already explained 
will give the 
f(9)-f(a«) + f(S«)-&c. =2£ (C) 
and if f(0) = A cos 9 -|- B cos 2 9 + C cos 30 -f- &c. a similar 
course will produce the equation 
f(d) + f( 2 «)+f(3#) + &c. = -if(o) (D) 
If a function is developeable in even powers of x, then its 
second function is developeable in the same manner, and so 
are all its higher functions ; therefore if f x and f are two func- 
tions developeable in even powers of x, such that 
fr = A 1 + BV-f CV+ &c. 
f x f ”jt = A + C.z 4 + &c. 
Then (A) will become 
Jii 2 li * 3 2i cxCC. — ■ 
= AS "7F + + &c - + J K ( E ) 
These theorems marked (A), (B), (C), and (D), although 
they possess a very great degree of generality, are not entirely 
without restriction; it appears at first sight that they are 
applicable to all functions which have the prescribed condi- 
tion of being expansible in even powers of the variable ; such 
was my opinion of them when I first discovered them ; but 
