An exact solution is obtained for the motion of the surface of a uniform elastic half-space due to the application at a depth H below the surface of a concentrated vertical force. The time-variation of the applied force is assumed to be represented by the Heaviside unit function. The solution for the horizontal and vertical components of displacement cast in the form of single integrals over a fixed range, and these have been evaluated on the electronic computer of the Weizmann Institute (WEIZAC). The assumed source emits both S waves and P waves. Beyond a distance r1 from the epicenter, which is equal to H/√ in the case λ = μ, the original S wave is converted on reaching the surface into a diffracted SP wave traveling along the surface. At large ranges, the SP phase is more pronounced than the P phase. The S phase is marked by a finite jump for r<r1, and by a logarithmic infinity for r>r1. The coefficient of the logarithmic term is zero both at r=r1 and at large ranges, having a sharp maximum at r=1.004r1. There is no Rayleigh wave at r<r1. At large ranges (r/H≫1) the solution, as a function of the reduced time τ = ct/R, approaches the form of the solution for the surface pulse.

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vibrating gaseous column [fibreS, raises their temperature and spreads out from place to place through the air and nearby bodies; but as this diffusion and irradiation occur with extreme slowness relative to the speed of the vibrations, one may suppose without sensible error that, during the period of one vibration, the amount of heat between two neighboring molecules remains the same." 5 The argument consists of a premise concerning the rate at which heat is lost, to the surrounding air and nearby bodies, and the conclusion drawn therefrom. The fallacy, if it exists, must be found in this premise.The inverse of "Laplace's fallacy" is the suggestion that his premise is erroneous because it is incomplete. Specifically, it is suggested that there can be no validity in any argument for a trend to isothermal compressions with lower frequencies, because the effect of heat conduction upon vibrations in an infinite medium is a trend to isothermal compressions with higher frequencies. This inverse fallacy can be found, explicitly or implicitly, in much of the recent literature) .ø-8 A summary of the earlier literature, which is now subject to criticism for supporting "Laplace's fallacy," may be illuminating. Stokes ø analyzed the effect of heat radiation upon the speed of an infinite plane wave of sound in a gas, neglecting heat conduction and viscosity. He also considered sound waves in a gas contained in a solid tube. His conclusions support Laplace's argument. Rayleigh TM presents a good summary of Stokes' analysis. Rayleigh TM also presents an analysis for the effect of heat conduction upon the speed of an infinite plane wave of sound in a gas, neglecting heat radiation and viscosity. The approximate results of this are, within the range of validity of the approximation involved, in agreement with more modern work on the same subject. Indeed Rayleigh's exact characteristic equation n can be shown to be identical in form to that presented as Eq. (9) by Deresiewicz/ when the latter is specialized to harmonic waves and the elastic coefficients are eliminated in favor of the isothermal and adiabatic velocities.

Using an extension of the method of Pekeris for S states, eigenvalues and eigenfunctions have been obtained for the 2 1 P, 2 3 P, 3 X P, and 3 3 P states of helium. The fine-structure splittings of the 2 3 P and 3 3 P states have been computed, including the a 3 quantum electrodynamic correction and the singlet-triplet correction. Determinants up to order n -220 were solved, and when the results were extrapolated to n -» oo good agreement was obtained with recent accurate measurements of the fine-structure splittings, substantiating the correctness of the a z terms.

The grand partition function of polyelectrolyte molecules, considered as linear Ising chains, is studied with respect to certain symmetry properties of these systems. The degree of ionization and the buffering capacity are shown to possess a symmetry with respect to the point of half neutralization. Equations for the pH, the degree of ionization, and the buffering capacity are derived explicitly for a model in which only the interactions between first- and second-neighbors are taken into account. The results are compared with those of the theory of first-neighbor interactions, and graphs of titration and buffering-capacity curves are given. The properties of the maxima and minima occurring in the buffering-capacity curves are discussed. It is found that the form of buffering-capacity curve provides information on the order of non-negligible interactions.