With the large necessities of the physicist and the growing require ments of the engineer, it is inevitable that the great majority of our students of calculus should need to use their mathematics readily and vigorously rather than with hesitation and rigor. Hence, although due attention has been paid to modern questions of rigor, the chief desire has been to confirm and to extend the student's working knowledge of those great algorisms of mathematics which are naturally associated with the calculus. That the compositor should have set vigor where rigor was written, might appear more amusing were it not for the suggested antithesis that there may be many who set rigor where vigor should be. As I have had practically no assistance with either the manuscript or the proofs, I cannot expect that so large a work shall be free from errors; I can only have faith that such errors as occur may not prove seriously troublesome. To spend upon this book so much time and energy which could have been reserved with keener pleasure for vari ous fields of research would have been too great a sacrifice, had it not been for the hope that I might accomplish something which should be of material assistance in solving one of the most difficult problems of mathematical instruction, that of advanced calculus.