Nearly six years ago, I presented to the Society two Memoirs on the subject of Differentiation, with fractional indices. The method which I adopted to extend the signification of a differential coefficient consisted in assuming that the function , which enters into the value of the coefficient deduced from a particular hypothesis, is limited only by the definition . This generalization appears to be perfectly satisfactory, and promises to offer, if not the only, certainly the best extension of the Differential Calculus. Considering the length of the interval which has elapsed since the publication of my former Memoirs, it is remarkable that so little addition has been made to our knowledge of this branch of analysis. With the exception of one or two papers in Liouville's Journal, and a few remarks by Professor De Morgan, in his Treatise on the Differential Calculus (pp. 598–600), I am not aware that anything has been written on this subject since that time. Seeing, therefore, that others are not willing to enter on this very promising field, I consider it not improper that I should make known a number of extensions of this science to which I have been subsequently led, many of which have been in my possession a considerable time.