1 
Mr. Babbage's essay towards 
Problem II. 
/ Given the same equation 
+ (*00 =+ {»£, fty ) 
Suppose one particular solution of this equation is known, 
let it b ef(x,y), 
then take \f/ (x,y) = q>f (x,y), <p being perfectly arbitrary 
and the given equation becomes 
= <pf(*x,fiy) 
which is evidently satisfied since/ (#,y) =/(«!, py) by the 
hypothesis. 
Ex. 1. Let $ (x,y) = 
one particular solution of this equation is f(x,y) = x hg ‘ y 
. hence the general solution is 
Ex. 2. Given the equation $ (x,y) = $ ( x n ,y n ) a particular 
case is/ (x,y) = hence the general solution is 
(!^) 
Ex. 3. Given the equation t]y (x, y ) = ^ (x n ,y m ) 
In order to get a particular case let us put 
f( x *y) = logdx -f-a log. 2 y 
by substituting this value we shall find that it is a particular 
solution of the equation, if a = — , 
hence the general solution of the equation is 
+ ( x >y)=<p (log.*# — i^ 1 og. a ^) = ? A°g. 
or changing the value of <p it becomes 
, , X ( (log. x) l0 S- m \ 
log. X 
'log. n 
(log.^y) log. m' 
