404 Mr. Babbage’s essay towards 
find /, so as to satisfy the equation 
fx—fccx ( a ) 
take any particular value 
Find/, so as to satisfy the equation 
1 
//*=//«£ *==/// i b ) 
1 1 1 
for it is known from (a) that 
/=/* 
take some particular value, and determine/ from the equation 
2 
fff* = Iff Pyx =fffyx> 
2 1 2 1 2 1 
continue this as far as 
ff (n) 
n n— 1 n n — 1 
then being any arbitrary function 
<p {//■ •//*} 
L n n — 1 1 
will satisfy the conditions of the problem. 
As an example, let it be required to find a function which 
shall not change, when for x we substitute x } — x , or ^== 
here a, x = — x, and & x = r = L. 
' \/x z —l 
(a) becomes f (x) =/ ( —x), 
hence/a: = q> [x, — a: J as a particular case, take/a: = x\ 
then ( b ) becomes 
/ X’ =f (/3 x)-- =/ 
1 1 I 
whose solution is fx — <p {a:, j 
