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The Bullard—Gellman formalism is applied to investigate the existence of convergent solutions for steady kinematic dynamos. It is found that the solutions for the Bullard—Gellman dynamo, as well as for Lilley’s modification of it, do not converge. Convergent solutions have been found for a class of spherical convective cells which would be stationary in a perfect fluid in the absence of rotation and of the magnetic field. By calibrating the theoretical magnetic dipole so as to fit the observed value at the Earth’s surface, one can find a dynamo in the above class which also matches the observed equatorial magnetic dipoles. There is a dynamo which has a rate of total ohmic dissipation of only 1.8 x 1016 erg s-1 for an assumed electrical conductivity of 3 x 10~6 e.m.u.'f This is one thousandth the rate of tidal dissipation, and one hundred thousandth the rate of heat outflow from the surface of the Earth. The required velocities are of the order of 10~3 cm s_1, and the average magnetic energy density is 4 erg cm-3. The internal structure of the magnetic field in this model shows a dynamo mechanism situated in the outer part of the liquid core and is thus insensitive to possible rigidity of the material in the * inner core.

An asymptotic theory is developed for the long-period bodily tides in an Earth model having a liquid core. The yielding inside the core is found to be different in the case of a stable density stratification from the case of an unstable stratification. In the latter case, a boundary layer is formed in which the stress decreases exponentially with depth below the core surface, the scale length of the exponent being proportional to the frequency. In the limit of vanishing frequency the stress tends to zero through most of the liquid core, except near the boundary layer at the surface, where it grows to a finite value. In case of a stable stratification, the stress oscillates with depth below the surface of the core with a wavelength which is proportional to frequency. An infinite number of 'core oscillations' with indefinitely increasing periods exist in a liquid core with stable stratification, but in the case of an unstable stratification, none exist above the fundamental spheroidal oscillation (53.7 min) for n — 2. The assertion made that a liquid core must be in neutral equilibrium is not true. The displacements and stresses within a liquid core in long-period tidal yielding are determinate, even in the static limit, and are not arbitrary. Love numbers are derived for uniformly stable, neutral, and unstable liquid cores, as well as for a model with a rigid inner core.

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Phys. Rev. 154, 40 (1967). 43J. M. Peck, Phys. Rev. 154,~" . (1967). 44At 1000'K, the Boltzmann factor for the seventh vibrational state of H2' is 5x10, compared with 5xl0 for the state v =2. Hence, if the electronic-coupling crossing-point mechanism introduced by Bates were significant in the DR of H2'+e, the mean cross section for the process would have to be at least six orders of magnitude larger than that of the noncrossing vibronic mechanism discussed here. As the results will show, this would imply that the cross section for the Bates mechanism wouldhaveto be at least 10 cm, for states with v~7, or that the rate coefficient for DR of H2' in high vibrational states would be of order 4 x 10 . This is between one and two orders of magnitude larger than any rate coefficient previously attributed to DR by the Bates mechanism.f values have been computed for the transitions m S-n P, m = 1-5, n= 2-5 and m S-n P, m, n= 2-5 for members of the helium isoelectronic sequence up to Z=10. The agreement between the results obtained using the dipole length, and velocity formulas, together with the convergence of the results as an increasing number of terms are included in the expansions of the wave functions, indicate that the values obtained are accurate to within 1% or better for the large majority of the transitions. Various authors have computed f values for transitions between S and P states of helium and of heliumlike atoms using different types of approximate wave functions. ' The wide variation between the results of the different calculations show the computed f values to be particularly sensitive to the wave function employed. Thus in order to ob-tain reasonably reliable f values it is necessary to employ wave functions of high accuracy. %e have previously' obtained accurate wave functions for the states 1'S and n S, n S, n'P, n P, n=2-5, for members of the helium isoelectronic sequence up to Z =10, and have therefore been able to carry out a systematic calculation of the f values for 886 SCHIF F, P EKERIS, AND AC CAD transitions between these states. The methods used to obtain the wave functions have been described elsewhere, ' ' and we shall only give a brief outline here. The nonrelativistic Schrodinger wave equation for a two-electron atomwith an infinitely heavy nucleus is solved by assuming a solution i.n the form of a series expansion possessing the appropriate symmetry. Each solution then gives an approximation to the nonrelativistic wave function for one of the states with the given symmetry, while the corresponding eigenvalue E gives an approximation to the energy of this state. To obtain the total energy, the contributions from the relativistic effects and a correction for the finite mass of the nucleus have to be added. The nonrelativistic wave function obtained is, however, accurate enough for computing the f values. The dependence of the wave function on the angular coordinates of the two electrons is determined by considerations of symmetry.Thus it remains to determine the dependence on the thre...