i88 Mr. Babbage's essay towards 
hence either of the following equations may be solved gene- 
rally, 
*:(*>>) =>H lb. 0) 
y x J 
= ( b ) 
the solution of the former is (x,y) = <P ( xy),and that of the 
latter is 4/ (x,y) — (x r ~ 1 y n ~ *) 
In (a) pu tn=r=z=~, then the solution of i|/ ( x, y ) = 
W xy , v/ xy ) is tj/ (x,y) = <p (xy) 
In a similar manner it may be found that the solution of 
+ (*.*) = + (r- T) 
is 4'(x,y) = <p (j) 
As another example take the equation 
t (x,y)= ^ S~y) 
a particular solution is ^ (x,y) — 2 xy +y \ hence the gene- 
rai solution is 4 (a:, y ) == <p ( 2 xy -}-y 2 ) , but we may find ano- 
ther particular solution of this equation which is totally diffe- 
rent from the former, and by combining the two we shall 
obtain a much more general solution. The equation (6) will 
coincide with the one under consideration, if we make n= — y 
and r = ~, then we shall have for another particular solution 
j (•£> y ) — hence a very general solution of the equation 
$( x >y) — 4 " (7^, V 2xy ] j is 
= §^=5} 
remaining perfectly arbitrary. 
In the equation of this Problem it may happen that 
