220 
Mr. Babbage's essay towards 
Our enquiries must therefore be directed to this point, and it 
will be found that f(x,y) = a — x —y has the required pro- 
perties, and is a particular solution of the given equation : 
hence the general solution is 
4 ,(.v,y) — <p‘ (a — (px — <py) 
There are many other particular cases which fulfil the same 
condition, such as 
these give the general solutions 
\~x—y 
i — bxy 
^ (*>y) = and 't ( x >y) = V 
Problem XXII. 
Given the equation 
Using the same substitution employed in the last Problem., 
this equation becomes 
(p 1 / 2 ’ 1 (<p x, <py ) . (p 1 / 3 ’ 1 ((px, (py ) = xy 
and putting <p x for x, and (p y for y we have 
$j 2 * x {x,y) • § f I, 2 ( x >y) = ( P Q y 
which becomes identical, if j 2 > *(x ,y) = x a nd/ r > 2 (x,y)—y 
consequently all the solutions of the last Proolem also solve 
this. 
Problem XXIII. 
Given the equation r 
F [x, -i; 1 ’ 2 (x } y)] ~ FjiJ; 2 ’ 1 (cr,y),y| 
this equation may be solved by the same artifice as the two 
last, assuming t \ (x,y) (p* f ((p x, <py) we have 
f [x, $f l >*{(px,<?y)} =F{ <p'f 2 > 1 <Py),y] 
