the calculus of j 'unctions. 401 
this subtracted from the former, leaves 
% (^) ± % (e~ y ) — 0. 
Let us first consider the upper sign, then a particular solution of 
% (■') = — % ( £ ~ y )> 
is % 00 — (log- y) 2n + l . 
If we take the lower sign, then a solution of 
% (&0 = % 
is % (y) = (1 og.y) tn . 
From these considerations it appears, that the general solu- 
tions of the given equations are 
+ y=Jy + ? 
and t| ,y=fy + <p {(log..)’)**} 
according as the upper or under sign is used. 
The equation just solved was not constructed as an example 
to this particular rule, but is selected because it has actually 
occurred. It is used by Mr. Herschel in the Philosophical 
Transactions, 1814, for the purpose of assigning the sums of 
several very curious series. He there observes, that when 
the upper sign Is used Zn L ( 1 +y), and when the lower 
takes place 2w + l L ( 1 -{-y, are particular solutions, these may 
therefore be generalized by the introduction of an arbitrary 
function 
If -J- tyz—y = a rational function ofy containing only even 
powers, 
or if 4* z y — 4* £ ~ y = a rational function of y containing only odd 
powers, 
they admit of the following solutions, in the first case, let 
^ e y -{- 4* £~~ y = a + # y 4 + ay 4 -{- &c. -}- a y in 
012 n 
3 F 
MDCCCXV. 
