i$4 1 Mr. Babbage's essay towards 
If therefore we know one particular solution of the original 
equation, and also one or two particular solutions of the other 
equation, we may deduce the general solution of the Problem, 
Ex. Let ty(x,y) = (-f j’*!' 
in this cas ef(x,y ) =/(y, x ) and two particular solutions are 
X £ 
f (x,y) = xy and f(x,y) — ^ -f y also a particular solution 
X r* 
of the given equation is/ (x,y) = hence its general solu- 
tion is 
— y f (x+y,xy) 
Problem VI. 
Given the equation 
x,y) = A[x,y) {a{x,y),l3 (iv,y)) + B (x,y) 
Suppose we are acquainted with one particular solution 
which satisfies the equation and let it be/ (#,y ), then assume 
4 O-.y) — f(x,y) + <p (x,y) 
and making this substitution the equation becomes 
/ ( X, y ) + ? ( x, y ) = A. ( x ,y )f ( a ( x, y ), /3 ( x, y ) ) +A ( x, y ) 
& (x,y)) + B (x,y) 
Substracting from this the particular solution 
/( *> y ) = A (. x,y )f ( * ( x, y ) , /3 (x ,y) ) + B (. x,y ) 
there remains 
rp{x,y') — A{x,y) <p(* (x,y), (i (x,y)) 
an equation which may be solved by the preceding Problem. 
The same substitution is applicable to the more general equa- 
tion 
o=4/R,y )+ a (*,>■) («(*>y )>/* ( x >y)) -b B C*by) ^ (*{*>y)A x >y))+ 
i i 
&c. K (x,y ) 
