OF LINCOLN'S INN, BARRISTER AT LAW; FELLOW OF THE CAMBRIDGE PHILOSOPHICAL VOLUME II. CAMBRIDGE: DEIGHTON, BELL AND CO. LONDON: GEORGE BELL AND SONS. 1888 [All Rights reserved.] PREFACE. THE second volume of this Treatise deals with the more advanced portions of Hydrodynamics, including the motion of viscous liquids to which the last four chapters have been devoted. It commences with a chapter on Harmonic Analysis, in which a variety of functions which frequently occur in physical investigations are considered. The most exhaustive work on this subject is the German Treatise on Kugelfunctionen by Heine, of which considerable use has been made, especially in the first twenty pages of this chapter. The remainder of the chapter which relates to Toroidal Functions, is taken from Mr Hicks' papers in the Philosophical Transactions for 1881 and 1884. The notation Jm (x) for an ordinary Bessel's Function of degree m is well established, and the second solution of Bessel's equation, which is not however so frequently required, may be conveniently denoted by Y(); but there is another class of functions also of considerable importance, which constitute the two solutions of the equation which is obtained by changing x into x in Bessel's equation. The notation for these functions does not appear to be so well established, many English writers employing the symbols Jm (x) and Ym (x), whilst German writers often employ the symbol K(x) in the place of Y (x). But as it appears to me that the employment of an imaginary argument in the case of functions which may always be treated as real quantities, creates unnecessary complexity, I have ventured to introduce a new notation, and have denoted these functions by the symbols Im (x) and Km (x) respectively. m The portions of Chapter XIV. which relate to the vibrations of a circular vortex and to linked vortices, have been taken with slight modifications from a paper by Professor J. J. Thomson in the Philosophical Transactions for 1882, and from the Treatise on the Motion of Vortex Rings by the same author, to which the Adams' Prize was adjudged in 1882. The latter portion of this chapter has been derived from Mr Hicks' papers on vortex rings in the Philosophical Transactions for 1884 and 1885. It is however necessary to point out, that the period equation obtained by Mr Hicks for determining the fluted vibrations of a circular vortex, does not agree with that obtained by myself, and consequently there is an important difference in the results connected with the stability of the vortex. I am however indebted to Mr A. E. H. Love, for having examined and verified the analysis of §§ 326—340, and I therefore trust that the results which are put forward are the correct ones. In the Chapter on Waves, I have made considerable use of Prof. Greenhill's Article on Waves in the American Journal of Mathematics, Vol. IX., which contains an exhaustive discussion of most of the principal problems of interest. The Chapter on the Tides is confined exclusively to the dynamical theories which have been proposed as an explanation of tidal phenomena, and is principally derived from the investigations of the late Astronomer Royal and Professor G. H. Darwin. The reduction of tidal observations, together with a variety of questions relating to the practical portion of the subject, are very fully treated in Professor Darwin's Article on Tides in the Encyclopaedia Britannica. Although nearly forty years have elapsed since the publication of Prof. Stokes' paper "On the Effects of the Internal Friction of Fluids on Pendulums," it is remarkable that very little progress has been made with respect to the solution of problems connected with the motion of solid bodies in a viscous liquid. The complete solutions for a sphere and a right circular cylinder moving in a viscous liquid of unlimited extent under the action of given forces, have not yet been obtained; and no problem involving the motion of two solids appears to have ever been attempted; neither have any general equations analogous to Lagrange's equations been discovered, by means of which the motion of one or more solids in a viscous liquid may be obtained, without going through the troublesome process of calculating the components of the force and couple exerted by the liquid on each solid. The difficulties of the subject are undoubtedly great, but it is hoped that before the termination of the present century, substantial progress will be made. I have in conclusion to express my obligations to Professor Greenhill for having read the proof sheets; to Mr A. E. H. Love for having examined the analysis of §§ 326-340, and for having read the proof sheets of the last four chapters; and to Professor J. J. Thomson and Professor G. H. Darwin for permission to make free use of their investigations on Vortex Rings and Laplace's Theory of the Tides respectively. UNITED UNIVERSITY CLUB, November 1888. |