III. On the calculus of functions

One of the first efforts toward the formation of the calculus of functions is due to Laplace, whose solution of the functional equation of the first order, by means of two equations in finite differences, is well known. Functional equations were afterwards treated systematically by Mr. Babbage ; his memoirs were published in the Transactions of this Society, and there is some account of them in Professor Boole’s Treatise on the Calculus of Finite Differences. A very important functional equation was solved by Poisson in his memoirs on Electricity; which suggested to me the investigations I have now the honour to lay before the Society.