the calculus exjunctions. 189 
u(x 3 y) does not contain x nor @ (x, y) contain y, it then be- 
comes $ (a?, y) = ^ («y, fix) 
This is the case when xj/ (x,y) is required to be a symme- 
trical function of x andy, the equation would then become 
■*/(.*, y) = Wy> x ) 
two particular solutions are/ (x,y ) = xy and f (x, y) = x +y, 
1 
hence the general solution of the equation is 
^(x,y) — <p {xy,x+y) 
Though these solutions may with propriety be termed general 
because they contain an arbitrary function, yet I am by no 
means inclined to think them the most general of which the 
questions admit, possibly we ought to except the two last equa- 
tions, though I shall afterwards show that the solution of an 
equation of the form § (x,y ) == t}/ (ax, fiy) may contain any 
number of known functions within the arbitrary one. 
Problem IV. 
Given the equation 
l\>(x,y) =r$(*(x,y),GXx,y)) 
Assume as before ip (x,y) =- <p \f[x,y) ,/ [x, y]), then the 
x 
equation will become 
Q {f( x >y)<.fi x >y)) =<?{ fH x <y), ft ( x ,y) )/(* i x >y),i K x >f ) } 
In order to render this equation identical, determine/ and/ 
x 
from the two equations 
f( x ,y) =/(“ i x > y)>P ( x >y)) s ndf(x,y ),==/(<*, (x,y), (i (x,y)) 
X I 
putting in the first of these a (x,y) for x and (3 (1 v,y) fory 
we find 
/.( ? i x ,y)> ft ( x J)) —/{<*{* [x,y), ft [x,y)), ft («( x >y),ft ( x ,y ) ) } 
=J( x >y) (0 
I 
C C 2 
