stability relative to this special class. Moreover the problem can be extended, within the framework of our general method, by considering obstacles whose contours are composed of a finite number of analytic arcs with shock waves originating at their points of intersection.It is evident that instability at V, or local instability, is sufficient to insure instability in the large. Hence, the above result on instability gives the complete answer to the problem of determining the conditions for instability of shock lines attached to the vertex V of an obstacle whose contour is an analytic curve. Since at most two shock angles a at V are mathematically possible the shock line which actually occurs and which corresponds to the shock line experimentally observed must therefore be the one whose inclination a lies in the interval ao(M) < a < A(M). This may be accepted as sufficient evidence for the stability (local or in the large) of shock lines with inclination a in the interval ao(M) < a < #(M) by those not interested in an existence-theoretic treatment of the problem.l Prepared under Navy Contract N6onr-180, Task Order V, with Indiana University.' The derivation of these relations and other results mentioned in this note are contained in several papers which we expect to publish later in the Journal of Mathematics and Physics under the following titles: "Calculation of the Curvatures of Attached Shock Waves"; "The Consistency Relations for Shock Waves"; and "The Distribution of Singular Shock Directions." The fundamental notion of statistical mean values in fluid mechanics was first introduced by Reynolds. His most important contributions were the definition of the mean values for the so-called Reynolds' stresses and the recognition of the analogy between the transfer of momentum, heat and matter in the turbulent motion.In the decades following Reynolds' discoveries, the turbulence theory was directed toward finding semi-empirical laws for the mean motion by methods loaned from the kinetic theory of gases. Prandtl's ideas on momentum transfer and Taylor's suggestions concerning vorticity transfer belonged to.the most important contributions of this period. I believe that my formulation of the problem by the application of the similarty principle has the merit to be more general and independent of the methods 530-PROC. N. A. S.

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I do not believe that one could connect justly the name of Gibbs with practical applications of applied mathematics, for his main interest was certainly centered on basic conceptions of mathematical physics. Nevertheless, for example, his beautiful work on graphical methods in thermodynamics is a brilliant example of the presentation of theoretical relations in a form which appeals to the engineer. This lecture is intended as an effort to improve the convergence between the viewpoints of mathematics and engineering. Thus, I feel it is not inappropriate to dedicate it to the memory of Josiah Willard Gibbs. Engineering mathematics is generally considered as a collection of mathematical methods adapted for the solution of relatively simple problems. These problems often might require lengthy numerical calculations or graphical constructions, but supposedly can be worked out without the use of advanced methods of mathematical analysis. This description was perhaps correct some decades ago ; today a large group of scientific workers is engaged in applying various methods of classical and modern analysis to problems in electrical, civil, mechanical, aeronautical and also chemical engineering. It is not possible to give an exhaustive list of all types of problems which require the applications of advanced analytical methods. In the following table merely some of the most important engineering problems and the mathematical concepts and methods involved in their treatment are indicated : ASSOCIATED TOPICS OF ENGINEERING AND MATHEMATICS Mathematical topics Vector algebra, systems of linear equations. Tensors and matrices. Algebraic equations. Ordinary differential equations with given initial conditions. Elementary operational calculus. Ordinary differential equations and their boundary problems. Eigenvalues and eigenfunctions. Expansion in or-Engineering problems Engine dynamics, vibration of systems with a finite number of degrees of freedom, Rotating electric machinery. Equilibrium, buckling and harmonic vibrations of beams. Critical frequencies and speeds. One-dimensional problems

The stress wave caused by a longitudinal impact at the end of a cylindrical bar has been analyzed in the case where the impact velocity is large enough to produce plastic strain. The theory gives a method for computing the stress distribution along the bar at any instant during impact. It is shown that for a given material, there is a critical impact velocity such that when subjected to a tension impact with a velocity higher than the critical, the material should break near the impacted end with negligible plastic strain. An experimental investigation was made concurrently with the theoretical study. Some of the most significant experimental results are presented in this paper.

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