The solution of a problem in classical dynamics can be described, as Hamilton showed in 1834, by its Principal Function. Considering for simplicity a conservative problem with one degree of freedom, let the co-ordinate at the instant t be q, and let the Lagrangean function be L. Let Q be the value of the co-ordinate at a previous instant T. Let the quantity , after the integration has been performed, be expressed in terms of (q, Q, t — T), and let the function thus obtained, which is Hamilton's Principal Function, be denoted by W.

The calculation of Van der Waals forces has acquired considerable interest recently through the work of Buckingham, Knipp and others (Buckingham, 1937; Knipp, 1939). In these papers the interaction energy between two atoms is expressed as a power series in i/R, where R is the nuclear separation, and the various terms in this series are known as dipole-dipole, dipole-quadrupole, quadrupole-quadrupole, etc… interactions. In most cases only approximate values are obtainable for the coefficients in this series, though for two hydrogen atoms in their ground states, Pauling and Beach (1935) have determined the magnitudes correct to about I in 106. In this paper we discuss the simplest possible problem of this nature, i.e. the force between a bare proton and a normal unexcited hydrogen atom. We shall show that a rigorous determination of the coefficients in the power series can be made.

With the help of a natural generalisation of the invariant scalar product for two spinor functions the invariant Fourier transformation of a spinor function can be defined, apart from a normalising factor. Assuming this factor as unity, the Fourier transformation of the solutions of Dirac's wave equation and its reciprocal are derived. The construction of reciprocal spinor functions leads to a transcendental equation for µ = ab/ħ which differs from that of the scalar case; but its roots are very similar to the latter.

Professor E. T. Whittaker has recently discovered a Third Quantum-Mechanical Principal Function R(q, Q, t - T) and has worked out the theory of this function in detail when the Hamiltonian is

By using the Sturm-Liouville theory of linear differential equations and the properties of Green's function, it is shown that the function is an elementary solution of the adjoint of the Schrodinger wave equation associated with the Hamiltonian H.

It is pointed out that the modified Planck constant ħ arises solely from the commutation relation and may, from the analytical view-point, be any constant, real or complex. In particular, if ħ = i, the use of an algebra with the commutation relation leads to an elementary solution of the real equation of parabolic type

It is shown that the elementary solution Γ(x, ξ; t – τ) of the equation

behaves, as t → τ + O, in very much the same manner as the elementary solution of the equation of heat.

10. Generating functions and bilinear generating functions (of the type of Mehler's celebrated formula) are known to be of great importance in the formal theory of orthogonal sequences. The present paper contains analogous formulae for a number of continuous orthogonal systems as well as “mixed” systems (which have a point spectrum as well as a continuous one). Four systems of the hypergeometric type have been selected as examples which are thought to be of some importance because of their presenting themselves in certain problems of Mathematical Physics.

My thanks are due to the Carnegie Trust for the Universities of Scotland for grants towards the printing of this paper and my paper in Proceedings, vol. lx, no. 26, 1940.

An examination of the Ben Nevis barometric records shows that, when the non-periodic convex variation is eliminated, the clear-day barometric curve differs but little from the normal curve obtained from all days. The results prove that the clear-day excess found by Buchan and Omond is mainly (though not entirely) due to bias in the selection of data.

In this paper the equations of the meson are treated in the same manner as the Dirac equation in a previous paper (K. Fuchs, 1940, Part IV). We use the formulae developed by Kemmer (1939), with a small modification, introduced so that the set of Maxwell's equations for the electro-magnetic field is obtained as a special case.

A quasi-field is denned by the postulates of a commutative algebraic field, except that the distributive law a(b + c) = ab + ac is replaced by a(b1+ … +bn)=ab1+ … +abn for a fixed integer n.

The properties of quasi-fields are investigated. The study of their ideals is reduced to the study of the ideals of a certain type of ring. A particular quasi-field is constructed formally by means of polynomial domains modulo a natural number, with addition specially defined.

Quasi-fields are connected with multiple fields—another generalisation of the conception of a commutative field, in which a fixed number of elements (> 2) co-operate symmetrically in the formation of any sum or product.

During the past four years I have been developing a new method of measuring the velocity of light. I had intended to defer publication until the method was perfected, but progress is becoming increasingly difficult owing to war conditions. So I give here an account of the progress already made. I am certain the method has a great future before it. There is a notice of it in a letter to Nature (Houstoun, 1938), but except for this no description of the method has yet been given.