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UNIVERSITY OF ASfRONOMYUBRARV JEr %ibris L5 BR AR Y OF THE ASTRONOMICAL SOCIETY FACiFJC 241 HYDEODYNAMIC S . Sonton: C. J. CLAY AND SONS, CAMBRIDGE UNIVEKSITY PEESS WAREHOUSE, AVE MARIA LANE. laggoto: 263, ARGYLE STREET. ILetpjig: F. A. BROCKHAUS. fo lorfc: MACMILLAN AND CO. HYDRODYNAMICS BY HORACE LAMB, M.A., F.R.S. PBOFESSOB OF MATHEMATICS IN THE OWENS COLLEGE, VICTORIA UNIVERSITY, MANCHESTER; FORMERLY FELLOW OF TRINITY COLLEGE, CAMBRIDGE. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1895 [All Rights reserved.] ASffiONOMYLIBRARY Cambridge: PRINTED BY J. & C. F. CLAY, AT THE UNIVERSITY PRESS. ASTRONOMY PREFACE. rilHIS book may be regarded as a second edition of a "Treatise -- on the Mathematical Theory of the Motion of Fluids," published in 1879, but the additions and alterations are so ex tensive that it has been thought proper to make a change in the title. I haveattempted toframe a connected accountofthe principal theorems and methods of the science, and of such of the more important applications as admit of being presented within a moderate compass. It is hoped that all investigations of funda mental importance will be found to have been given with sufficient detail, but in matters of secondary or illustrative interest I have often condensed the argument, or merely stated results, leaving the full working out to the reader. In making a selection of the subjects to be treated I have been guided by considerations ofphysical interest. Longanalytical investigations, leading to results which cannot be interpreted, have as far as possible been avoided. Considerable but, it is hoped, not excessive space has been devoted to the theory of waves of various kinds, and to the subject of viscosity. On the other hand, some readers may be disappointed to find that the theory ofisolated vortices is still given much in the form in which it was. left by the earlier researches of von Helrnholtz and Lord Kelvin, and that little reference is made to the subsequent investigations of J. J. Thomson, W. M. Hicks, and others, in this field. The omission has been made with reluctance, and can be justified only on the ground that the investigations in question L. b M6772O1 VI PREFACE. derive most oftheir interest from their bearing on kinetic theories of matter, which seem to lie outside the province ofa treatise like the present. I have ventured,in one important particular,to make a serious innovation in the established notation of the subject, by reversing the sign of the velocity-potential. This step has been taken not without hesitation, andwas only finallydecided upon when I found that it had the countenance of friends whose judgment I could trust but the physical interpretation of the function, and the ; far-reachinganalogywith the magnetic potential, are both so much improved by the change that its adoption appeared to be, sooner or later, inevitable. I have endeavoured, throughout the book, to attribute to their proper authors the more important steps in the development of the subject. That this is not always an easy matter is shewn by the fact that it has occasionally been found necessary to modify references given in the former treatise, and generally accepted as correct. I trust, therefore, that any errors of ascription which remain will be viewed with indulgence. It may be well, moreover, to warn the reader, once for all, that I have allowed myself a free hand in dealing with the materials at my disposal, and that the reference in the footnote must not always be taken to imply that the method of the original author has been closely followed in the text. I will confess, indeed, that my ambition has been not merely to produce a text-book giving a faithful record of the present state of the science, with its achievements and its imperfections, but, if possible, to carry it a step further here and there, and at all events bythe due coordina tion of results already obtained to lighten in some degree the labours of future investigators. I shall be glad if I have at least succeeded in conveying to my readers some of the fascination which the subject has exerted on so long a line of distinguished writers. In the present subject, perhaps more than in anyother depart ment of mathematical physics, there is room for Poinsots warning