the calculus of functions. 237 
in order to determine the constants b and b, substitute this 
x 
expression in the given equation and it will be found that 
b 
- — --- = — bp therefore 
2 p+l J 
p + I „ —~p 
vJat = — b px b x 
c s 
in which there still remains one arbitrary constant, 
It is observable that both these solutions contain one con- 
stant. Let us suppose this to be changed into an arbitrary 
function of x, and let us determine what conditions it must 
be subject to, that it may satisfy the Problem: taking the 
second example we have 
-P 
'tyx = (x 
and the equation becomes 
-P 
p + 1 
■px ) <px 
- P px P ')<p-J=(x 
[x 
from this equation <px must be determined ( the method of 
doing which will appear in a subsequent Problem). If this 
solution contains an arbitrary constant, the same process may 
be again repeated. We may thus continue deducing one 
solution from another as long as we can solve the differential 
equations to which they give rise, but still these will only be 
particular solutions. 
Problem XXXIV. 
Given the equation 
* ux =n? 
and d*x. = x, put for x successively ax, a 2 x, . . »P— * x then 
we have 
d n -i/x 
