39° Mr. Babbage’s essay towards 
It is this inverse method with respect to functions, which I 
at present propose to consider. 
If an unknown quantity as x, be given by means of an equa- 
tion, it becomes a question how to determine its value ; simi- 
larly if an unknown function as 4/, be given by means of any 
functional equation, it is required to assign its form. In the 
first case, it is quantity which is to be determined ; in the second, 
it is the form assumed by quantity, that becomes the subject 
of investigation. In the one case, the various powers of the 
unknown quantity enter into the equation; in the other, the 
different orders of the function are concerned. 
Before I proceed, it will be proper to explain the mean- 
ing of the order of a functional equation, and likewise to 
indicate the notation made use of ; «, jQ, <y, &c. are known 
functional characteristics; %, \p, are unknown ones. 
If in any function as ij/x, instead of x, the original function be 
substituted, it becomes 4* x or 'k 2 x • this is called the second 
function of x . If the process be repeated, the result is \f 2 \^x 
or 4 3 •£> the third function of x ; and similarly 4'* x, denotes the 
n 1h function of x. Suppose 
= a -f- x 
then ^x — a+a-{-x = x 
and generally y x = na + x 
A functional equation is said to be of the first order, when it 
contains only the first function of the unknown quantity ; as, 
for instance, 
^ ax x-^ x x” ~o 
( tyx -f vj/ ■j) s — ax + .r 2 = o. 
