the calculus of functions. 
1 95 
Problem VII. 
Given the functions 
a. (x,y),a {x,y)i * {®,y) &c. 
i a 
/3 fay) &c - 
I 2 . 
Required the nature of the function ^ (%>y) such that it 
shall not alter its form by the simultaneous substitution of 
« {x ,y), ]Q (x,y) for x and y, and generally that it shall re- 
main the same when for x andy are respectively substituted 
any of the functions denoted by « (a?, y) and fi (x,y). The 
n n 
conditions which determine may be thus expressed 
$ (*r,y) = + (*(v*y),P (#,y)) = + (f {*>?' )>& (*>y ) ) = 
Assume ^ ( x,y) = q> [/(#».?)>/ } ( x ) 
then from the first condition we have 
<? ==?> {/(«(*. y)>Q( x >y))J{« { x >y)>K x J))} 
this will be satisfied by making 
f{x,y)=f (a (< v,y),fi(oc,y)) and f(x 9 y) =f{«(x,y),(l(x,y)) 
these are two particular solutions of the first equation. 
The second condition \s -ty (x,y) =-ty (u(x 9 y), @ (a?,y)) 
i i 
which becomes 
<p (J(i,y),f(/v,yy) = <P (/(« (a?,y), £ (x,y)) , f (ct {x,y ) , /3 (a?,y)) ( 2 ) 
v 1 1 * * * * 
where jf and /are known functions; make 
2 
/O /3 (a?,y)) = K (x,y) and/(« £(r,y)):==*K(a?,y) 
s X XI I 
K and ’K are therefore also known functions. 
MDCCCXVI. 
D d 
