the calculus of functions. 213 
In the same manner it may be shown, that if we take the 
f h simultaneous function, and then put x fory, the result will 
be the same as the p th function of cpx, or expressed in symbols 
it is 
t p >P(x,y) = (p p (x) [y=x] 
Now since this equation is identical when y is equal to x, it 
will remain so when any other quantity as v is put for x , if 
the same quantity is also put for y , therefore 
v) = (pt* (v) 
now let v=fy {x,y) this equation becomes 
l/ p ’ p ($(x 9 y) 9 ^ (x,y)) =..<p* + (x,y) 
but the right side of this equation is nothing more than the 
p-\-ith simultaneous function of (x,y), consequently 
If now in the equation of the Problem we substitute the seve- 
ral values thus formed of the simultaneous functions, we shall 
have 
(x,y), <p^(x,y),f X P (x,y), . . <p f ~' + (x,y) J =o 
and putting % for $ (x,y) we have 
F [z,(pz,(p* z , . . <p ^~ l z j =0 
which is a functional equation of one variable, and may be 
solved by the methods of the first Part. The form of cp being 
thus ascertained, we have for determining $ (x,y) the equa- 
tion 
i\j(x,y)=cpx [y = x] 
or expressed in words $ (x,y ) may be any function of x and 
y which becomes equal to <px when y is equal to x . 
Ff 2 
