The method of expansions, developed by Lagrange in his Théorie des Fonctions Analytiques, has been for many years almost exclusively adopted in this University for the demonstration of the formula of the differential calculus. The great name of the originator of this system gave a certain permanency to the method which probably it would not have possessed had it emanated from one less illustrious. Not to insist upon the doubts which have been thrown by several recent writers on the validity of conclusions deduced from the comparison of infinite series, it is certain that the absence of

any notion of limits in the algebraical theory of derived functions, gives rise to an entire want of homogeneity between its fundamental conceptions and those which present themselves in its most interesting applications. Within the last few years an endeavour to re-establish the system of limits, has been made by several elementary writers in France, among whom may be mentioned Moigno, Duhamel, and Cournot, and by Professors De Morgan and O'Brien in England. From my own strong conviction of the marked advantage which the method of limits possesses over that of derived functions, both abstractedly and in its applications, and, trusting to the