200 
Mr. Babbage’s essay towards 
Problem X. 
Given the equation \|/ 2> 2 (oc,y) =; a 
In this case the second simultaneous function of a? and y is 
constant. The first solution which presents itself is ^ Qc,y') = 
x—y + A, then we shall find 
■*t / ’ y') — ((# — y ~f* A) — (oc — y -j- A) -{- A ~ A = a 
therefore A -=a and one particular solution is 
4 1 ( oc,y) = oc — y -f- a 
This may be rendered more general, nearly in the same 
manner as the last Problem ; thus let ^ (oc,y) = (oc—y) (p (a?,y) + a 
then 
4 2 ’ ’O, y) = IQc—y f (x>y) + <0 — (x — y <p (x, y ) + a)] x 
<p y — y <p ( [x,y ) + a,x — y <p(x,y~) -f al -f a — a 
Another particular solution which readily occurs is 
a A 
^ (%>y) = A A this gives ^ 2 > 2 (oc,y) = A — — = A =a 
• A A. 
therefore A = a and a particular solution is 
4< oc,y) (oc,y~) = A 
this readily points out another general solution, let 
^ (a?,y)==A(p ( A ] hence i}A 2 (oc, y) = A <p 
A<p(i)=tf 
make A = and the general solution is 
Kav) = 577J ? 
From the combination of the two preceding solutions we 
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