the calculus of functions, 
other more general ones are 
211 
4>{x,y) 
(x z + .y g )p(i) 
x 
2 X <p 
I 
,n—i 
> and4/(x’,j') = < 
2 xy<p (i) 
(x -f y) <p 
x I J 
n • — i 
Problem XV. 
Required the solution of the equation 
— - n—p, n—p 
V = F* (x,y) 
(f'A-, 
In (6) we have 
»n,n. -v n 
V (x 5 jy) = <p 
(#, y) _j_ \ 
consequently 
n—p, n—p 
4/ (x,y) and substituting this value, we shall find 
p n -p(*&2\ + i\ 
* * 
n — p( a '('L , 2)n -\- 1 
<3 (±y) n 
n, n 
p n — p, n — p 
= <p 4> (*>y) 
make <p p u = Fu and find the value cp, then the general solu- 
/*(£»_y) # . A 
tion of the equation is 4 (x,y ) = ) if « (l, i) = 
/3 (i, i). ___ __ 
£x. l. Let 4 4 ’ 4 (^ y) = 2 (x,y ) | 
here <p 2 u = Fu = u 2 and <pu = i/ 2 
f * 
therefore 4 (*, y ) = j 
* if « (i, i) = /3 (i, i) 
more particular solutions are 
i/ a 
x+y ^ 1 1\ 1 f -v. .A— { * % +;*y + / yp 
2 tp 
-?(i) ) and<J»(^,^) = 
3X 
.£ 
MDCCCXVI 
F f 
