PREFACE. 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 formulæ 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 valuable opinion of many writers of the present day on the comparative merits of the two systems, I have been induced to enter upon this treatise. My object has been to present to the English student a work from which he may acquire a thorough and systematic knowledge of the abstract theory of limits, and of its applications to certain branches of coordinate geometry. How far I may have succeeded in this attempt, and of the liability to failure in a work of such nature I am fully sensible, will be determined by the judgment of the reader. a In the composition of this work, which was commenced in the April of this year, I have derived great assistance from Moigno's Leçons de Calcul Différentiel et de Calcul Intégral, Duhamel's Cours d'Analyse, and Cournot's Théorie des Fonctions: from the last of which treatises I have made considerable extracts in Chapter VII. on the Development of Functions. Cambridge, November, 1845. Definition of a differential coefficient Differentiation of the sum of a function and a constant Differentiation of the product of a function and a constant 15 Differentiation of the product of two functions 16 Differentiation of the ratio of two functions Differentiation of the product of any number of functions 20 Differentiation of a function of two functions Differentiation of a function of any number of functions of a 22 Differentiation of an implicit function of a single variable. General theory of the differentiation of implicit functions of a Total differentiation of a function of functions of independent Partial differentiation of an explicit function of n +r variables, 27 Partial differentiation of an implicit function of two independent Differential coefficient of x' 31 Differential coefficient of loga x 32 Differential coefficient of a* 33 Differential coefficient of sin x 34 Differential coefficient of cos x 35 Differential coefficient of tan x 36 Differential coefficient of cot x 37 Differential coefficient of sec x Differential coefficient of cosec x 39 Differential coefficient of sin ' 40 Differential coefficient of cos x 41 Differential coefficient of tan ̄'x 42 Differential coefficient of cot-'x 43 Differential coefficient of sec-1x 44 Differential coefficient of cosec-1x 45 Differentiation of simple functions of y with regard to x Order of partial differentiations indifferent Successive differentiation of an explicit function of two func- Successive differentiation of an implicit function of a single Successive total differentials Successive differentiation of an explicit function of three vari- |