the calculus of functions 
solution of our equation is 
207 
a 15 (f>y) n 
, &c. 1 
f J 
a. lx, y) 
where the numbers m, n, p , &c. are not confined to integers. 
Another solution may be found in the following manner : 
let *(*>>> = *(pg 3 ) = K 
and determine % so that % (x, y) = % (x, y) when y is made 
1 1 
equal to a?, then taking the second simultaneous functions on 
both sides* we have 
*’"(*> j) = * i)) = f(0 
a general solution of the equation in question is therefore 
^ (*> y) — ff) § 
a 
this solution depends on that of the equation 
% y) = x (# , y) D = ^3 
£ 
which belongs to a class of equations we shall speak of here- 
after. 
Let us now return to the consideration of equation (4) it is 
4 < 2 , 2 (x,y) = <P 
i tibj} r a ( * (x -' y) « yt -,b 1 
1) (x,y) m ) J ) 
for n put n - {-1 and for m put n , then it becomes 
* 2 ~ ( ^ } = ^{ 1 
take the third simultaneous function then 
fa{x, 3»L Li 
0 
