4°8 
Mr. Babbage’s essay towards 
On the number of arbitrary functions mtroduced into the complete 
When from a functional equation of the first order, we 
determine the form of the unknown function, one or more 
constant quantities are generally introduced ; these as I have 
shown in a preceding Problem, may be changed into arbi- 
trary functions of the unknown quantity which fulfil certain 
prescribed conditions. 
A. question naturally arises as to the number of these arbi- 
trary functions, and how many any given equation admits of 
in its most general solution. 
The train of reasoning usually made use of to prove, that 
a differential equation of the n th order, requires in its complete 
integral n , arbitrary constants may be pursued on the present 
occasion, though from several reasons, it would perhaps be 
desirable to have a proof resting on a different principle ; as 
I have not been successful in discovering any other, I shall 
give the only one I am at present possessed of. 
Let -ty x = F| x, a, a, . . a l 
for x, put any number of known functions, as ax, fix , . . vx s 
the results will be 
solution of a functional equation. 
2 
n 
(°) 
( 1 ) 
( 2 ) 
&C. 
&C. 
(») 
