PREFACE. ز The following attempt to set forth in a systematic and connected form the present state of the theory of the Motion of Fluids, had its origin in a course of lectures delivered in Trinity College, Cambridge, in 1874, when the need for a treatise on the subject was strongly impressed on my mind. Various circumstances have retarded the completion of the work in a form fit for the press; but as the delay has enabled me to incorporate the results of several important recent investigations, and altogether to render the volume less inadequate to its purpose than it would otherwise have been, this is hardly matter for regret. I have endeavoured, throughout the book, to attribute to their proper authors the various steps in the development of the subject. The list of Memoirs and Treatises at the end of the book has no pretensions to completeness, and as it is to a great extent based on MS. notes which I have no present means of verifying, some of the references may possibly be inexact. I trust however that the list may, in spite of these drawbacks, be of service to the student who wishes to consult the original authorities. I am under great obligations to my friends Mr H. M. Taylor and Mr W. D. Niven of Trinity College for their kindness in correcting the proof-sheets and in generally supervising the passage of the work through the press. HORACE LAMB. ADELAIDE, May 16, 1879. CONTENTS. The Eulerian form of the equations. Dynamical equations, equa- tion of continuity, surface conditions Second method. Flow of matter, momentum, energy Impulsive generation of motion The Lagrangian form of the equations. Dynamical equations, Comparison of the Eulerian and Lagrangian methods. 18-21. . 22–27. Velocity-Potential. Lagrange's Theorem. Equi-potential sur- faces. Physical interpretation of velocity potential Analysis of the motion of a fluid element Irrotational motion in simply-connected spaces Particular case of a liquid. Properties of the velocity potential Irrotational motion in such spaces. 41, 42. 43—52. 53–59. 38 47 Irrotational motion. Kinetic energy Relations between the velocity potential and the stream-function . Digression on the theory of functions of a complex variable. Examples. Special cases of motion General formulæ for the velocity potential and the stream-function Motion of a circular cylinder in a liquid. Motion of translation of a solid through a liquid . Motion of rotation of a solid in a liquid Helmholtz's and Kirchhoff's solutions. Impact of a stream on a lamina. 89. 91 90–93. 94 . 99-104. Kinematical investigations Direct method. Application to a sphere. 110_112. Kinetic energy. Momenta 113, 114. Cases of steady motion . 115. Motion of a solid struck by an impulsive couple 116. Expressions for the kinetic energy in certain cases. 117. Motion of a solid of revolution in a particular case. 118. Stability of a body moving parallel to an axis of symmetry |