2 6g 
sums of several classes of infinite series. 
-M. = + _A_ + . E . ■ 4- &c. + K + Lx s 3 *+ Mx'3"+ 
3 2k v Zk > f k X Zk ‘ f k — Z X Zk ~~ Z ' ‘ ‘ ° ‘ 
- f yl ^ 5-r &c. — K — Lx* 4? - MjV— 
4 2 ^ 2A fk-2 x 2k—2 
Sc c. &c. 
&c. 
If vve add the vertical columns, we shall have on the left side 
of the equation the series 
1 f 
X 1 * l l U 
f ( 2 *) , f(3*) f(+») | } 
2 * “T “T | 
and the right side of the equation consists of three kinds of 
terms, those which contain negative powers of x, one which 
does not contain x, and the remaining ones which contain 
positive powers of x. With respect to these last, they are all 
of the form Oi s; { i 2? — 2’’+ 4®'+ & c * } ; an< ^ as the series 
which multiplies Oaf* is equal to zero, all those vertical columns 
which contain even powers of x will vanish : the term which 
is independent on x is 
K — K-j-K — K-f- &c. = -|K 
and those terms which contain negative powers of x, may be 
represented by the expression S All the vertical columns 
being summed, we shall have the equation 
f(x) f( 2 x) , f( 3x) 
jZ/c 2 2 k ‘ ^2 k 
&c - — AS ~fk + BS -fk=r 2 + 
+ + &c - + i K ( A ) 
As the operations by which we have arrived at this expres- 
sion have been given at length, it will be unnecessary to 
repeat them with the slight modifications which would be 
required for cases nearly similar. Thus, if we suppose the 
function fx developeable according to the odd powers of 
x, and if we divide both sides of the equation by x 2k+l and 
