268 Mr. Babbage’s new methods of investigating the 
provided we put r for x afrer the operation K is executed; 
that is, we have found the values of the series 
Ajj,* -j- A 2 je* 4* A x® 4"* &c. 
in the particular cases of x which satisfy the equation ctx — x. 
I shall now explain another method of deducing the sums 
of a variety of series, which comprehend amongst them all 
those which are contained in the former part of this paper ; 
it rests fundamentally on the following formulae, which have 
long been known : 
It is unnecessary to give proofs of these and other similar 
ones which have been frequently noticed, as they may be very 
easily demonstrated. 
Let fx be any function of x developeable in even powers 
of x, then f(x) = A Bx* 4 - Gz 4 + Dx°-{- &c. 
Divide both sides by x 2i , then it becomes 
For x , put successively lx, 2x, 3a?, 4/v, &c. and let the alter- 
nate series be taken negatively ; these being arranged under 
each other, we have 
Part II. 
0= i 6n — 2 Z ” + 3 * n — 4 2 ” + &c, 
4 — cos 0 — cos 20 -{- cos 3 9 — &c. 
4 = cos 0 + COS 20 4“ COS 3^ ”1“ &c. 
zkjlk ,zk zk 
2 X 2 X 
Zk — 2 x Zk~Z 
B 
B 
+ &c. + K + LxV+ M#V+ 
* — &c. — K — LxV— Mx*2* — 
