210 
Mr. Babbage’s essay towards 
Problem XIV. 
Given the equation 
r^(x,y)==Ft(x,y) 
In equation (6) we have 
fi—l /'“(f’jK+i' 
— — n 
/« (^y) n+1 \ 
P ut for 9 KW-K~) its value tp ( x , y) then we have 
+ (*>y) = <p n $(x,y)==¥^(x,y') 
determine <p from the equation <p n ~~ l v == Fu by Prob. 13. 
Part 1. and the general solution of the equation is 
+ <?’y) = *t?tep—) 
Ex. 1 . Let xp 3 » 3 {sc, y) = ^ (#» y) 
in this case ^ w ” x u — Fu becomes cp 2 v=u solutions of which are 
1 
, m 
<£11 = —-, <pv=a — v, (pv,=(a — v m y 
let a (fV^) M+1 = xr + y 3 and (3 (x, y) n = x 4- y then solutions 
-of the equation 3 (x, y)~-b (x,y) are 
4 / (x, y) = $(x,y) = a — j +y% 
+ y 
and 4/ (x, y) == [a — (£l±|?j w J 
Ex. 2. Let ^ n * n {x,y) = j + (x, y) |“ 
in this example = Fu becomes <p n ~ l v=v m and <pu=v 
its particular solutions are therefore 
1 
rn 
71—1 
( 
* 3 + / 
* + y 
n — 1 
x a + 2x y '—y ' 1 
n — 1 
