the calculus of functions. 187 
If n 2= m we have (iv,y) == <p (™~ ) as in the last example, 
and if m = — , we have the same solution as in the first. 
n 
In these equations the functions have contained the varia- 
bles separated ; but it may frequently happen, that they occur 
mixed as in the following Problems. 
Problem III. 
Given the equation 
Assume ip (x,y) = <?>(/ (*>y)>f ( x >y))> ® n d by making 
2 
this substitution the equation becomes 
<p {f{ x >y)J( x >y)}=<p {f{a{ x >y)>$( x >y))’J{“{ x >y)>£{ x >y )) } 
In order to render this equation identical, I determine/ and/ 
x 
from the two following equations : 
From these it appears, that/ and/ are merely two particular 
x. 
solutions of the original equation. If, therefore, we are ac- 
quainted with any, the general solution is 
^(x,y)=<p{f{x,y),f(x,y)) 
X 
If only one particular solution is known, the general one is 
4 (ar.y) =<pf(x,y) 
Ex. 1 . Let us examine in what cases we can find the general 
solution of the equation 
(x,y) = 1}/ ( x n y m , x k y r ) 
In order to obtain a particular solution, put *]/ (x,y)=x»y w 
and making this substitution, we shall find the following equa- 
tion of condition among the exponents. 
( 1 — • n ) (1 — r ) = km 
' C c 
MDCCCXVI. 
