sums of several classes of infinite series. 
f —4_ I f I +20 v/ 1 I X$s/— 1 7 -1 
c '~ ir ' { i+**o* + i+xw j I = °L(2). 
279 
or 
S(-i) 
X+ J 
7TIF = t or 
f == 7+0* i+z J 0* 4“ 1+3*6*' (4>4) 
The same series being summed by the theorem (A) in this 
paper gives the same result; but then the theorem alluded to 
only declares this to be the true value in case a certain series 
is finite, which series is 
4- 9 \ 1.2.3 + ^.1.2.34.5 4- 6V1.2..7 4- &c. 
but this series can only be finite when 9 is actually equal to 
zero : the method which I have explained in this paper points 
out that the equation 
2 i+&» 1+2*0* 4 * r+ 3*6* 
can only be depended on when 9 = 0, in that case it is known 
to be correct. I have already stated that the reason why the 
value of the series so found is incorrect, is that the series 
_0*(l*_ 2 *-J- & c .) + 6 4 (i — 2 4 + &c.) — 9 \i 6 — 2 a -f &c.) + &c. 
has been neglected because the coefficient of each term is 
zero. I shall now proceed to investigate the sum of this series, 
and shall prove that it is equal to a finite function of 9 : let 
y = C 2 (lV— 2V*-f &c.) + c 4 (iV — 2V*+ &c.) + &c. 
then y is equal to the sum of the series whose value we are 
seeking ; if c— 9 ^/ — 1 and x = 0, differentiate y twice relative 
to x, and multiply by c 3 , and we find 
C tt ~ = c\ lV — 2V V + &c.) 4- c\ lV— 2Y*4~ &c.) 4 - &c. 
Hence the equation for determining y is 
c * if ~y = — c X V£X — 2*^4- &c.) 
And the value of y is 
— &c. 
O+O 3 
