*Theoretical Hydrodynamics* by L. M. (Louis-Melville) Milne-Thomson (1891–1974) is, along with Horace Lamb’s older book [L], and Landau and Lifshitz [LL], one of the classic reference works in fluid mechanics. The word hydrodynamics is a somewhat older term used to describe what in more contemporary parlance would be called fluid dynamics and comprises both fluids and gasses. The book first appeared in 1938 and went through an unabridged fifth edition, on which this Dover reprint is based. As is the case with many Dover reprints, the book is also available in electronic form free on Archive.org.

Most of the book emphasizes mathematical techniques/approaches to understand inviscid flow. The first chapter, *Bernoulli’s Equation*, is mainly introductory, somewhat historical in flavor and tends to be less technical than most of the subsequent chapters. It states, derives and explores some of the consequences of Daniel Bernoulli’s famous equation. Chapter 3, *Equations of Motion* covers the general equations of motion and develops Euler’s equations of fluid motion. The more general Navier-Stokes equations are developed much later in chapter 21, *Viscosity*, along with more advanced topics of waves, vortex motion and boundary layers.

There is considerable attention paid to complex variable and conformal mapping methods. The idea here of course is that the velocity field of two-dimensional incompressible irrotational fluids may be modeled by functions \(\overline{f(z)}\) where \(f(z)\) is analytic. The fifth chapter amounts to a concise but introductory course in complex variables. In fact ten of the twenty three chapters employ complex variable methods to some degree.

This book is one of the great classics in fluid mechanics, but if it has one shortcoming, it is probably the lack of attention paid to conceptual and physical thinking behind the models. The book focuses on the mathematical methods of fluid dynamics, such as, for example, uniqueness theorems, and the motivating physics is sometimes lacking.

While the book is encyclopedic and a must have for specialists in fluids, apart from the first introductory chapter, it may not be the best starting point for beginners whether they be undergraduate or graduate students or professionals wanting to learn more about the subject. Some books of more recent vintage a bit more accessible include the book by Acheson [A], suitable for advanced undergraduates and beginning graduate students. The books by Chorin and Marsden [CM], Batchelor [B], and White [W] are more advanced but also highly regarded in their own right as well as helpful companion texts.

The book contains an extensive collection of exercises, many taken from the Cambridge Tripos, the University of London’s M.Sc. examination and the Royal Naval College.

**References**

[A] D.J. Acheson, *Elementary Fluid Dynamics*, Oxford Applied Mathematics and Computing Science Series, Oxford University Press, 1990

[CM] Alexandre Chorin and Jerrold E. Marsden, *A Mathematical Introduction to Fluid Mechanics*, Springer-Verlag 1992

[L] Horace Lamb, *Hydrodynamics*, Cambridge University Press, 1879 (With later editions and Dover reprints.)

[LL] L.D. Landau and E.M. Lifshitz, *Fluid Mechanics*, 2nd edition, Elesvier, 1987

[B] G.K. Batchelor, *An Introduction To Fluid Dynamics*, Cambridge Mathematical Library, Cambridge University Press, 2000

[W] Frank M. White, *Viscous Fluid Flow*, 3rd edition, McGraw-Hill, 2006

Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.