the calculus of Junctions. 
Problem XXXVIII. 
Given the equation 
^ {Xi y )= ^) 
where a y=y. 
For y put ay and the equation becomes 
differentiate this relative to x, then we have 
dp (x, ay) 
dx 
d*P(x,y) 
dx x 
this substituted in the original equation, gives 
4 * (*> y) — 
which is a partial differential equation, whose solution is 
4 ‘{x,y) =c"tpy +7W 
1 
(py and py being two arbitrary functions ofy, so constituted as 
£ 
to fulfil the original equation. These may thus be deter- 
mined, since 
+ ( *>y) =£ x <t>y + £ X( py 
we have 
dp (x, ay) 
dx 
£* (pay — £ (pay 
and, since these two quantities must be equal, we have the 
following equations 
cpy s= (pay and (py = — (pay 
s 1 
the former of these is easily satisfied by putting for < py any 
symmetrical function of y and ay; and a particular solution of 
the latter is 
<py — {—y + *y)c 
I 
and since this solution contains an arbitrary constant, it may 
Kk 2 
