We consider an ideal problem of adsorption of single and double particles upon a solid surface which has its sites of accommodation regularly arranged, and by comparing the equilibrium properties obtained by Bethe's method with the ordinary statistical formulae, we obtain approximate expressions for:(1) g(N, n, X), the number of ways of arranging n particles upon N sites of a lattice so that the number of neighbouring sites occupied by the particles is X.(2) g2(N, n, X), the number of ways of arranging n double particles upon N sites so that each of the double particles takes up two adjacent sites and the number of neighbouring sites occupied by two different particles is X.Both these expressions are found to agree with the exact values when the N sites lie on a straight line. When we use the first expression to construct the configurational partition functions of certain physical assemblies and expand them in powers of 1/kT, they are found to agree with the corresponding rigorous expressions as far as (1/kT)3, which is the highest power which we can find rigorously at present. With the help of the first expression, formal equations for superlattice formation in an alloy with the composition 1: 1 and equations for the separation into phases of regular liquids are given. Lastly we show that atoms and molecules in a regular liquid may dissociate or recombine suddenly accompanied by a latent heat. This is a new cooperative phenomenon, which may bear some resemblance to the melting process between the solid and liquid states.

The theory of stability of the flow of a viscous, electrically conducting fluid between rotating cylinders in the presence of an axial magnetic field is extended to the case where the cylinders are permeable and the primary flow includes a radial component. Numerical results pertaining to the stationary axially symmetric modes are presented, and the asymptotic stability behaviour for large values of the radial Reynolds number is derived.

The relativistic field theories of elementary particles are extended to cases where the field equations are derived from Lagrangians containing all derivatives of the field quantities. Expressions for the current, the energy-momentum tensor, the angular-momentum tensor, and the symmetrized energy-momentum tensor are given. When the field interacts with an electromagnetic field, we introduce a subtraction procedure, by which all the above expressions are made gauge-invariant. The Hamiltonian formulation of the equations of motion in a gauge-invariant form is also given.After considering the Lagrangian L as a scalar in a general relativity transformation and thus a function of gμν and their derivatives, the functional derivative ofwith respect to gμν (x) at a point where the space time is flat is worked out. It is shown that this differs from the symmetrized energy-momentum tensor given in the above sections by a term which vanishes when certain operators Sij are antisymmetrical or when the Lagrangian contains the first derivatives of the field quantities only and whose divergence to either μ or ν vanishes.

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In considering the onset of instability of an electrically conducting fluid between rotating permeable perfectly conducting cylinders in an applied axial magnetic field, it is found that oscillatory axially symmetric modes occur at large values of Hartmann number, in addition to the usual stationary modes. Results are presented showing the effect of the oscillatory modes on the criterion of onset of instability.The asymptotic behaviour of the stability criterion is considered in the limit of very large radial Reynolds number, and also in the limit where both the radial Reynolds number and the Hartmann number are large.

The transition of the onset of instability from stationary modes to oscillatory modes for an incompressible, conducting Couette flow between two coaxial, perfectly conducting, non-permeable, rotating cylinders under the influence of an axially applied magnetic field is considered. Results for three cases are reported. These pertain to flow between (1) a rotating inner wall with a stationary outer wall, (2) counterrotating walls, and (3) corotating walls. It is found that, for high values of the Hartmann number, there may exist some range of convective wavenumbers for which neither of the two lowest modes of axisymmetrical disturbances will become stationary. Within this range, the neutral stability curve is determined by a complex-conjugate pair of oscillatory axisymmetrical modes of equal stability. The oscillatory modes may, in fact, become more critical than the stationary modes. It is demonstrated that the approximation of replacing the angular speed by its average value, combined with the assumption of a narrow gap between the cylindrical walls, eliminates the oscillatory axisymmetrical modes.

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