C 2 5 3 
III. On the developement of exponential functions ; together with 
several new theorems relating to finite differences. By John 
Frederick W. Herschel, Esq. F. R. S. 
Read December 14, 1815. 
In the year 1772, Lagrange, in a Memoir, published among 
those of the Berlin Academy, announced those celebrated 
theorems expressing the connection between simple exponen- 
tial indices, and those of differentiation and integration. The 
demonstration of those theorems, although it escaped their 
illustrious discoverer, has been since accomplished by many 
analysts, and in a great variety of ways. Laplace set the 
first example in two Memoirs presented to the Academy of 
Sciences,* and may be supposed in the course of these 
researches, to have caught the first hint of the Calcul des 
Fonctions Generatrices with which they are so intimately 
connected ; as, after an interval of two years, another demon- 
stration of them, drawn solely from the principles of that 
calculus appeared, together with the calculus itself, in the 
memoirs of the Academy. This demonstration, involving, 
however, the passage from finite to infinite, is therefore 
(although preferable perhaps in a systematic arrangement, 
where all is made to flow from one fundamental principle) 
less elegant ; not on account of any confusion of ideas, or 
want of evidence ; but, because the ideas of finite and infinite, 
as such, are extraneous to symbolic language, and, if we 
* Mem. des Savans Etrangers, J773. p. 535,— Mem. del’Acad, 1772.P. 102, 
mdcccxvi. E 
