the calculus of functions. 1 93 
the second and third class may be reduced to those of the 
first, I shall therefore always consider functions of the first 
order as simultaneous ones, and omit the indices, which if 
supplied, would be [» ** &c. 
1, 1 
To transform (3 (.r,y)) into a function whose 
index is * put a (x,fi (x, y) ) = <y (a:,y) then 
*> 4 
2 0 /s (?,y ) ) = 4< {“( x ’y)’K x ’y)') 
and similarly if /3 (a (x,y), /3 (r,y) )=y (x, y) we should have 
I) 1 
I, I 
i!; 2 ,I («(a;,y),/3 (x,y)) = ty 1 * *(* (x,y), 7 (*,y) ) 
and generally whatever be the number of variables a similar 
transformation might be effected. 
Problem V. 
Required the solution of the equation. 
4' {x>y) = A (*,y) $ (* (x,y), (3 (x,y)) 
Assume (x,y)=f(x,y) <p [f(x ? y),f(x,y)J and sub- 
stituting this in the given equation, we find 
J(x,y ) <p {f (*, y) ,f ( x, y ) } = A ( x, y)f ( * (x, y), Q (x, y ) ) 
This equation will be satisfied if we are acquainted with par- 
ticular solutions of the three following equations 
f{x,y) = A(x,y)f(*(x,y),l3(x,y)) 
J{x,y)=J{*{x,y),${x,y))™df(x,y)=f(x(x,y),Q(x,y)} 
a * 1 1 
the first of these is nothing more than the original equation. 
) 
