326 
Proceedings of the Royal Society 
way from any given function another set of functions, to which the 
name of first, second, third, &c., differential co- efficients are given. 
The corresponding problem here is to interpret the operating symbol, 
where fx may have any value whatever. This interpretation must 
be so made that it shall include the particular meanings already 
attached to the cases where fx is integral. This process of extension 
has been aptly called by De Morgan a case of the interpolation of 
forms, and there are difficulties in connection with it very like those 
that arise in the solution of functional equations or the inverse 
method of finite differences. The question was first raised by 
Leibnitz, and was treated successively by Euler, Laplace, Eourier, 
Liouville, Greatheed, Peacock, and Kelland. 
The laws of operation to be conserved are — 
D n (u + v) =D n u + D w ?;, 
D m D n u = D m+n u. 
The only question is : — What fundamental functions are we to select 
on which to base our calculus ? It appears that different systems 
arise according as we select our fundamental functions. Peacock 
starts with x m : Kelland, following Liouville and others, starts 
with e mx as the ground function, and lays down the equation 
as the foundation of his system. 
By means of a definite integral he then deduces the general 
formula 
/dfY x - n _ ( - 1 ) n Jn + fx 
\dxj fa x n+n ’ 
where jn is a function like the gamma function, satisfying the 
equation 
In + 1 = n jn , 
but unlike it not restricted to positive values of n. 
This formula is, then, applied in a variety of particular cases, 
and is shown to be perfectly general provided certain conventions 
are adopted, and from it are derived working formulae convenient 
in different cases. 
