20 6 
Mr. Babbage’s essay towards 
then the equation is reduced to 
— i) 
If therefore we assume 
f a -f bx + cx z & c .q 
by cxy I 
*( x >y) = yh) <P < 
cy 
2 
<P C 1 ) ~ 1 * a + X t)x -f Sec. 
'by 'cxy 
> 
'ey 1 
a 
the original equation will be satisfied. 
I am inclined to think, that this solution is not the most 
general of which the Problem admits, even though the series 
were continued back, as it might, to negative powers of x 
and y. The two solutions which follow' are possibly more 
general, although on this point I am not certain. It would 
indeed be a very important step, if we could assign the number 
and nature of the arbitrary functions which enter into the com- 
plete solution of functional equations. 
Another solution of the equation $ ** z (j?, y) = a may be thus 
deduced, let 
^(x,y)= <p 
£C TA &c 
“ (X> y) ’ & <£• y) ’ 
i ■ m i — — n J 
= K 
then taking the second simultaneous function on both sides, 
it will be perceived by the construction of the second side of 
the equation, that 
« (** 0 V (WQ ftrf. 
« (n O’ B (i, i)> 
call the right side of the equation A, then a very general 
