Proceedings of the Royal Society of London/Volume 2/On the Developement of Exponential Functions together with several new Theorems relating to finite Differences
The subject here considered by Mr. Herschel relates to the celebrated theorems of Lagrange, expressing the connection between simple exponential indices and those of differentiation and of integration.
Since the theorems have been demonstrated by various subsequent analysts, as by Laplace, by Arbogast, and by Dr. Brinkley, the author takes them for granted; but observes that in their original form they are but abridged expressions of their meaning; and that in order to become practically useful, their exponential functions require further development.
And though this part of the subject has been treated with great ability by Dr. Brinkley, who has deduced formulae respecting the first of the two theorems far more simple than could have been expected from the complex nature of the subject; yet since his method, when applied to the second more general theorem, would lead to details of extreme complexity, Mr. Herschel has taken a different view of the subject ; and beginning with the more general theorem has arrived at a general formula, which he believes to have been hitherto wholly unnoticed, and which, when applied to certain particular cases treated of by Dr. Brinkley, affords precisely the same results.
But the mode in which this subject is treated, or even the results, were not of a nature to admit of being read in public.