Mr. Babbage’s essay towards 
418 
make fx equal to x, for in this case we at once determine ^x 
to be equal to x, which is not always true. 
The same observation may be made with respect to making 
/* x = x, for in this case y x = x, 4 ^ x = 4' x, y x — x, 
yxzs-fyx, and we in fact by assuming this value for / deter- 
mine \|/ from the equation 
F | X , 4/07, X , 4/07, 07, &C. |=0 
The same objection does not hold when we make f 3 x=zx, 
though this considerably limits the generality of the solution ; 
apparently the most eligible mode of determining/ is from the 
equation /»+* x^x, for in this case supposing we are acquaint- 
ed with a particular solution of ( a ) containing any number 
of arbitrary constants, such as 
<p 07 = A 1 07 , a, b, c, &c» j 
w@ may substitute for each of these constants an arbitrary 
function, such as 
% 1 07, fx,f* 07, .../ n 07j 
for it is evident this function does not change when we substi- 
tute f m x for 0 ?. 
But this form of / is not always correct, it may be incon- 
sistent with the original equation. In fact, the only value we 
can assign to /which shall not in some measure limit the 
generality of the Problem, is to suppose a particular solution 
of the given equation. 
As an example of this method take the following problem. 
D 
B C 
