Malecot(4) under certain conditions derived the formula for the covariance of the genotypic values of a quantitative character in two individuals AI and AII, which were related but not by direct descent. This generalized some results of Fisher (l). Kempthorne (2) extended the theory to multiple allelic systems with any degree of epistacy (i.e. interlocular genie interaction) but without linkage. He gave the formula Here is the item in the population variance which can be attributed to the interaction of additive gene effects at r loci and dominance gene effects at s loci. φ and φ′ are the coefficients of relation between the two individuals. The various assumptions normally included under random mating equilibrium were made, i.e. no selection, mutations, maternal effects or differential viability. Kempthorne (2), (3) gave two rather different proofs of this important result. His second proof was the more straightforward, but it was somewhat condensed.

The recurring controversy about clocks and relativity reveals the need for an examination of the meaning and methods of time measurement.

A simple method for constructing irreducible representations of the symmetric group is given. It is particularly suitable for use in quantum mechanical calculation of interactions in many particle systems, using electronic digital computers.

Several forms of the lower-bound variational method for the calculation of the eigenvalues in a wave-mechanical problem are considered, and compared; the particular case of the harmonic oscillator being chosen. All forms have certain unsatisfactory features, but some of them are considerably worse than others. One reason why calculations of lower bounds are in general less satisfactory than Ritz-type calculations of an upper bound is shown to be that whereas, in the presence of a scale factor, this latter wave-function satisfies the virial theorem, in none of the lower-bound wave-functions is this true. Similar calculations are reported for the ground state of the helium atom.

The calculation of the interaction between a point multipole moment and an external charge distribution is considered in elliptic coordinates. It is complicated by the singularity at the point moment. The results obtained are useful in calculating hyperfine interactions in diatomic molecules where the elliptic coordinates are the most natural system of coordinates. The results lead to an extension of Heine's expansion of (λ + μ)−1.

The effect of a twofold bond alternation on the bond orders and and on the π-electron contribution to the diamagnetic susceptibility of a cyclic even hydrocarbon polyene is investigated. Power series expansions and asymptotic expressions are given for and χπ in terms of the parameter t2 (0 ≤ t2 < 1) which characterizes the amount of bond alternation. It is found that bond alternation reduces both bond order and χπ strongly, the effect being more drastic the greater the bond alternation (t2→0).

Various authors, e.g. Sommerfeld (8), Bouwkamp(i), Groschwitz and Hönl(2), Müller and Westpfahl(7), have considered approximate solutions to the problem of the diffraction of a plane wave by a narrow perfectly conducting strip. If the electric vector is polarized parallel to the strip they find that the scattering coefficient, for normal incidence, is given by, where k is the wave number, 2b the width of the strip and γ(= 0·5572…) being Euler's constant. Any attempt to discuss the convergence of such a pseudo-power series requires a knowledge of the higher order terms and naturally leads to the question of what is the proper expansion parameter.

This paper is concerned with dual series equations of the form where is the associated Legendre function of degree n + m and order −m of the first kind, Hn is a known coefficient of n, f(θ) and g(θ) are given functions of the variable θ and the equations are to be solved for the unknown coefficients An. In hydrodynamics, elasticity and electromagnetism such equations occur in many problems involving a spherical or spheroidal boundary when different conditions are prescribed on different parts of the boundary. These dual series equations are analogous to the dual integral equations which arise in the application of Hankel transforms to problems involving plane or paraboloidal boundaries, the theory and applications of these latter equations having been discussed by several writers amongst whom are Sneddon, Noble and Lebedev(1), (2), (3). The main result of this paper is that the solution of the dual series equations (1·1) and (1·2) for m a positive integer or zero can be reduced to the solution of a Fredholm integral equation of the second kind in one independent variable. The restriction that m be a positive integer or zero covers those dual series equations which arise in three-dimensional boundary-value problems but excludes those which arise in two-dimensional ones where m is . These latter equations have, however, been discussed in a recent paper by Tranter (13).

The 3-dimensional potential due to a charged circle can be derived in various ways which lead to the same expression in terms of the toroidal coordinate σ through Legendre functions of s (= cosh σ). Applying the same methods to the magnetic potential of a circular current we find various results: (a) Legendre's equation yields no solution (§4·1). (b) The appropriate transformation from cylindrical to toroidal coordinates gives a Fourier series (equation (48)) the coefficients of which, though not Legendre functions, are related to them by the identity (§4·2) By this method we also derive expressions in elliptic integrals (equations (56), (58)). (c) Direct integration of the solid angle gives an alternative expression in elliptic integrals (equation (59)).

A study is made of the waves generated by a magnetic line source placed at the vertex of a wedge of high conductivity and arbitrary angle. The boundary-value problem is reduced to the solution of a difference equation and an exact solution obtained. The method is also applied to the case of dielectric coated wedges where the surface reactance and resistance are arbitrary, and the propagation of surface waves along such surfaces is considered briefly. The forms of the solution for large and small values of the surface impedance are obtained and show complete agreement with the known results available for a right-angled wedge and a plane.

The flow pattern past a thin wedge-like profile set at a small angle of attack in a gas flowing with high subsonic or sonic velocity is discussed within the order of the transonic approximation. In the model considered the flow has a stagnation point at the nose of the wedge and breaks away, with velocity equal to that of sound, from the shoulders. The velocity is subsonic throughout the whole field of flow. The solution of the boundary-value problem for the wedge in a channel is formulated as a pair of dual integral equations. The complete solution is given for the wedge in a free stream and the dimensions of the profile, together with the lift coefficient, are computed as functions of the transonic similarity parameter.

The effect of surface tension on the stability of two superposed fluids can be described in a universal way by a non-dimensional ‘surface tension number’ S which provides a measure of the relative importance of surface tension and viscosity. When both fluids extend to infinity, the problem can be reduced to the finding of the roots of a quartic equation. The character of these roots is first analysed so as to obtain all possible modes of stability or instability. Two illustrative cases are then considered in further detail: an unstable case for which the density of the lower fluid is zero and a stable case for which the density of the upper fluid is zero, the latter case corresponding to gravity waves. Finally, the variational principle derived by Chandrasekhar for problems of this type is critically discussed and it is shown to be of less usefulness than had been thought, especially in those cases where periodic modes exist.