214) 
Mr. Babbage's essay towards 
M 
Problem XVIII. 
Given the equation ^ (x, y) ~F (x,y) 
F (x,y) being such a function of x and y that it may be redu- 
cible to the form F y (x,y ) c 
Assume 4- = | 
then from eq. ( 7 ) we have 
i 
— m 
make 
« (*>>). 
— n 
o (x y ) — «y (-r, y) = v, and since y (a?, y) is given and 
a and [3 are indeterminate, this equation may be easily satisfied 
in an infinite number of ways ; put v for y (a?, y) our equation 
becomes 
1 
n — m 
- „ . \ _ ri—in 
| lilii? <pp v | = Fz> 
which contains only one variable and may be solved by the 
methods of the former part* 
Ex. 1 . Let 4 rp (». >) = [j ) 
Assume a (x,y) = (x+y)f [jj and /3 (x,y) = stj( 1 ) 
then it becomes 
<pPv — v 
and calling/ any particular solution of this equation we have 
S 
for the general one 
4 (x,y) = <i / ? 
