496 ON THE SUMMATION OF SERIES, 
when multiplied by its increment, shall be equal 
to a given function of vw. The determination of 
this function will be, frequently, a matter of con- 
siderable difficulty ; and consequently, can be 
accomplished, in finite terms, only in a few cases. 
F(a+1) b 
If we denote the direct operation of F@) 
F(x), so that “hey = F(x) =f (x), and the in- 
verse operation of f (xv) by (2), so that F (x) = 
F(x) ; a direct and inverse calculii of the fune- 
tions denoted by these symbols of operation may 
be readily established. 
The developement of these calculii, (which 
may, possibly, throw some light upon our present 
acquaintance with the doctrine of Algebraical 
equations, ) I shall reserve till some future oppor- 
tunity. Nor will the above idea respecting 
equations appear to be altogether visionary ; when 
we consider that Algebraical equations are al- 
ways formed by the continual product of factors. 
