If now the function y is an eigen function and its dependence upon X is explicitly known, this formula yields the normalization constant r(x)y2 dx immediately. Examples are the Jacobi polynomials or the functions of Ref. 2. In either case the functions y are expressed by hypergeometric series, the endpoints of the interval are singular points. The formula may even be useful in problems for which the eigen functions are determined by numerical integration for different values of X and subsequent interpolation. The hodograph transformation [1, 2] for plane compressible flow is too well known to need any discussion here, whilst its singularities have been very fully investigated in [3,4,5]. Briefly, it has been found that in cases where plane potential flow is impossible, the continuation of the solution in the hodograph plane may be regular but cannot be mapped into the physical solution. In any given hodograph solution there is no difficulty in deciding if a solution is mappable, for the condition of the vanishing of the Jacobian of the transformation is just that the level lines of the stream function in hodograph space should touch the fixed characteristics in the hodograph space. This criterion of course requires the calculation of all the level lines. Beyond this not much is known, but it has also been shown by various writers, particularly Friedrichs [6], that limit lines cannot appear initially in a certain sense at points inside the solutions. The purpose of this note is to show that the somewhat abstract results of Friedrichs, may be included in the simpler and rather more general statement, that for a wide class of hodograph solutions, if the boundary streamline can be mapped then so can the whole field.*Received June 15, 1951.

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Print) 1941-5990 (Online) Journal homepage: http://www.tandfonline.com/loi/tphm18 LX. Expansion in series of the exact solution for compressible flow past a circular or an elliptic cylinder A.R. Manwell To cite this article: A.R. Manwell (1945) LX. Expansion in series of the exact solution for compressible flow past a circular or an elliptic cylinder, The London, Edinburgh, View related articles On the Exact Solution for Compressible Flow past a Cylinder.499 elastic and viscous constants, since the nodes become ill-defined and are considerably displaced from the positions they would occupy in purely elastic systems. The shortest road to their elucidation in such a case is probably to plot a set of curves such as those of figs. 1 or 2 for a number of values of z., expressed as multiples of fi, and to compare the theoretical and practical distributions of amplitude, noting in particular the values at the best approximations to resonance, i. e. segmental delineation, that one can attain.The deduced values of N and ~ are shown beneath the figures. It may be objected that the physical characteristics of sols and gels should not be separated into rigidity and viscosity in the manner adopted in these experiments, that, in fact, " jelliness " is an indivisible property. Maxwell, however, regarded such systems as characterized in this way, the ratio of viscosity to elasticity being a dependent variable v characteriz ing the rate at which the system recovers equilibrium when displaced. When the liquid is subjected to simple harmonic forcing, the effective viscosity ~ should decrease from the " rest-value " V0 according to the formula 7=V o 1_t_p%2, so that the viscosity will effectively experience a rapid drop as p approaches 1/~. Evidence for these ideas cannot be derived from the present experiments, since in the systems used 1/r is a quantity probably much larger than the maximum frequency attainable. Summary.By subjecting a liquid having elasticity and viscosity to shearing forces of simple harmonic nature, it is shown how values of the two coefficients (of elasticity and viscosity) may be deduced by plotting the distribution of amplitude of oscillation throughout the system. The amplitude and the relative phase of motion are measured by means of hot-wire anemometers. Results are presented for oil, and for sols of cellulose acetate and gelatine of small concentration.

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