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The following is consequently the outline of the treatise. It consists of three parts; viz. Integral Calculus, commonly so called; the Calculus of Variations; and Differential Equations. The notion of a Definite Integral is stated in its fundamental and most comprehensive form; and the first four chapters are occupied with theorems, evaluations, and other properties of these Integrals. In the VIth and VIIth chapters, the Theory of Single Integration is applied to certain geometrical problems and to the theory of series. In the VIIIth chapter a large extension is given to the subject by means of Multiple Integration, the Elementary Theorems and the Theory of Transformation of Multiple Integrals occupying that chapter. In the three following chapters important applications of this theory to Geometry and to the Calculus of Probabilities are explained; and in the XIIth chapter various methods are discussed which mathematicians have devised for the Reduction of the Order of Integration.

In the course of the Calculus of Variations I have taken the opportunity of expounding at some length the properties of geodesic lines, and in an especial manner those of geodesics on an ellipsoid.

The third part in which Differential Equations, that is element-functions involving two or more dependent variables, are discussed is necessarily imperfect; the subject is surrounded with difficulties and is close on the present boundaries of our knowledge; I can do little else than exhibit such detached portions of it as have yielded to the powers of Analysis,

I am, as in the first Volume, under obligation to many friends for assistance and advice; to Professor Stokes of Pembroke College, Cambridge, to Mr. W. Spottiswoode, M. A., of Oxford, to Mr. H. J. S. Smith, Savilian Professor of Geometry, Oxford ; to Professor De Morgan, to M. Moigno, to M. Duhamel ; and to many others whose contributions are acknowledged in various parts of the Treatise. And I am also bound to express my sense of obligation to M. Liouville, and M. Crelle, on account of their valuable Journals.

The Chapters mark the salient divisions of the matter; the Articles are numbered continuously throughout the Volume, and their numerals are placed in the inner corners on the top of the pages. Bracketed numerals are also attached to the more important equations and are separate for each Chapter ; and reference is for the most part made to the numbers of the Article and of the equation.

The references throughout are made to the second edition of Vol. I.

11, St. Giles', OXFORD.

July 1st, 1865.

ANALYTICAL TABLE OF CONTENTS.

INTEGRAL CALCULUS.

CHAPTER 1.

THE THEORY OF DEFINITE AND INDEFINITE INTEGRATION.

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Art.
1. Integration, a process of summation .. .. ..
2. Differentiation, a process of disintegration .. ..
3. Examples of integration as a summatory process
4. The general form of a definite integral .. .. ..
5. Relation of the definite and indefinite integrals .. ..
6. The definite integral is independent of the mode of division..
7. Relation of symbols of differentiation and integration .. ..
8. Fundamental theorems of definite integrals .. .. ..
9. Examples of definite integrals determined from first principles

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SECTION 1.- Integration of Fundamental Algebraical Functions. 11. Integration of a" da .. .. .. .. .. .. 12. Integration of x-ldx .. .. 13. Examples in illustration .. .. .. .. ..

s dx da 14, 15. Integration of

of x2 + a2' =x2-a2 .. .. .. ..

- dx 16. Integration of hom and

ac" (a + bx) .. " 17. Examples in illustration .. ......

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