EDITOR'S NOTICE. THE first edition of this work, which appeared at the close of the year 1841, having been exhausted, a new edition is, under the sanction of the Proprietors, now presented to the public. The Editor has not attempted to make any alterations in the general arrangement of the treatise, but has confined himself to effecting such occasional changes in the details, as would probably have been thought desirable by the author had he lived to prepare another edition for the press. The last chapter, however, on the Comparison of Transcendents, offers an exception to these remarks, having been in a great measure rewritten: for the alterations in this department of the work, the Editor is indebted to the kindness of Mr Ellis, Fellow of Trinity College. WILLIAM WALTON. CAMBRIDGE, DIFFERENTIAL CALCULUS. CHAPTER I. * DIFFERENTIATION. Functions of One Variable. Ir u be an explicit function of a, which is of a complicated form, it may generally be reduced to the differentiation of simpler functions by means of the theorem y being some function of x, and u some function of y. This theorem may be extended to any number of functions, so that |