the calculus of functions* 
1 85 
then the given equation becomes 
<p (fr,fy) = 
% 1 
Determine/ and/ from the two equations 
s 
fx = fax and fy =/ 3 y 
z 1 
this may be effected by Prob. L or IL of my former Paper, 
take any particular solution and (p may remain perfectly arbi- 
trary ; then the general solution of the problem is 
+ {?,y) = <p t f x >fi ) 
Ex. 1. Given the equation ^(x,y) = 7) 
here we have/(x)=/( 1 — x), and a particular solution is/xrrrx 3 ; 
also/ (y) =/(“) and a particular case is/(y ) 
hence the general solution of the equation is 
= *(**. nr) 
<p being perfectly arbitrary. 
If we employ the general solutions of the equations 
/ ( x ) = / ( - x ) and/ ( y ) ==/(y ) » we shall sti11 onl y have one 
arbitrary function. In fact, the most general solution of the 
equation y ) = ^ yj with which I am at present 
acquainted is 
'K*..)') = f {— x > x >y> t } 
and this only involves one arbitrary function. 
