94 
Proceedings of the Royal Society 
as the complete integral, so we see that 
i 
y 
y 
= 
Various classes of cases in which this form is integrable are 
given, of which the following is one : — 
Let $=rj J Q, then the equation becomes integrable in the form 
7] 
2 , ~ + fJQ — 0 
> z + mrj + 1 ^ 
( 3 ), 
provided 
Q' 
p ^ + * 7W = m( ^ 
i.e.. 
€ -/Pdx 
~jw 
- m f € ' 
f Fdx c lx , 
The next subject treated is the effect of the alteration of sign 
of P or Q in (2). This is illustrated by the equation 
y" ±xy' ±y = 0 , 
which is integrable or at least reducible to quadratures for any of 
the four combinations of sign. 
The always integrable case where 
P = (C-a>)Q 
is next examined. 
Another portion of the investigation deals with certain infinite but 
convergent series, whose sums can always be expressed in terms of 
the integral of a linear differential equation of the second order. 
Consider, for instance, the expansion 
€ X = 
+ px +...•+■ 
1+V 
X 
+ 
+ . . 
= 2 P nX n , suppose. 
Obviously we have 
P n = p*P_ n =^ + 
From this at once 
•) n + 1 
\n 
+ 
P 
n + 2 
|1 n + 1 j2 \n 4- 2 
= P n _ i , whence P ra = (/ dp) n P ( 
dp 
• ( 5 )- 
