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The uniqueness of the solution can be proved as follows: Suppose that 4>i(x) and $z(x) are two solutions such that *i(x) = f(x) + ("" K(x, sfaWds,

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The velocity fields associated with a variety of flows which may be described by perturbations of the Blasius solution are considered. These are flows which, for example, because of localized mass transfer, involve the initial-value problem of boundary-layer theory, or which involve a variable ratio of the viscosity-density product, or finally which involve mass transfer. The perturbation solutions are presented so that in accord with the usual linearization procedures further applications for the determination of first-order effects can be readily made. In addition, each of these perturbations involves a common differential operator whose eigenfunctions form a complete orthogonal set. Accordingly, a procedure for systematically improving each perturbation solution to obtain higher-order effects by quadrature is presented. The results of applications in several cases are given and are compared to more accurate solutions where available.

REVIEWED BY W. R. DEAN 1 THIS BOOK gives an introduction to the hydrodynamics of inviscid fluids and is, after the preliminary chapters, mainly concentrated on two-dimensional motions. There is a full treatment of the general two-dimensional motion of an immersed cylinder with particular reference to the flow past a thin airfoil and past an infinite array of airfoils, and a full treatment of discontinuous motions and free surface flows. The analysis that is usually given of the stability of two-dimensional vortex systems is here significantly extended by some work with equations accurate to the second order of the arbitrary displacements, and interesting conclusions are drawn from this extension; this important section is based on work attributed to A. M. Liapunov.

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