THAN THE FIRST, CONTINUED. 
and finally 
y = //...Xdx n + c^- 1 + c^” -2 ...+ c n . (7). 
We shall hereafter shew that the first term in the second 
member may be replaced by a series of n single integrals. 
2dly, If the equation be of the form — Y, it is not gene 
rally integrable, but it is so in the case of n = 2. Thus there 
being given 
(Py 
we have 
and integrating 
Hence 
2 É^ = ovÉ 
dx dx ¿ dx ’ 
(!)= 2 STd, + a 
d £=(3jYdy+G)\ 
dx = 
x = 
(ZSYdy+C)*' 
dy 
(2/Ydy+Cy 
As a particular example, let = a 2 y. 
+ C. 
Here 
(2 Jd 2 y dy + C) 
= a 
+ 0' 
= /(ay+0)* 
= \ lo g \ a y + V(«y + O)} + O'. 
Case II. When the given equation merely expresses a 
relation between two consecutive differential coefficients. 
B. D. E. 14