It has been acknowledged for a long time that current quantum theory is incomplete. The difficulties and unanswered problems which have gradually become apparent during the development of the theory will not be discussed here, and only one aspect of the situation will be mentioned, that there seems to exist a large number of particles with different rest-masses, the numerical values of which demand a theoretical explanation. The experimental material has recently been greatly increased by the discovery of several kinds of mesons with different rest-masses.

1. Introduction.—One of the most important of Milne's discoveries is undoubtedly the significance of time-scale regraduations and, in particular, the relation between atomic t-time and gravitational τ-time. Although in Milne's work t-time is more fundamental than τ-time, this relationship is not inevitable, as was shown in an axiomatic development of cosmology given recently by the author. There the τ-scale was the more fundamental, and it was not found necessary to introduce Milne's t-scale. The object of the present paper is to discuss this primitive τ-scale still further, and to show how the t-scale may be introduced by means of an axiom. The unpublished work mentioned above is not required for this purpose because the cosmological models considered here were described in earlier papers (Walker, 1937, 1940 b). We also examine the various constants, absolute and conventional, which are connected with the different scales of time and length, and with different models.

Relativity is the study of matter in motion, and the basis of a theory of relativity can be either physical, mathematical, or logical. It is physical if some of the elementary objects and relations are concepts derived from the external world and if certain of their properties are assumed as physically obvious. If, however, the elementary objects, etc. are defined as mathematical symbols and relations, and if the subsequent theorems are mathematical deductions from these definitions, then the theory may be described as mathematical. Lastly, the basis of a theory is logical if certain terms are undefined—and clearly stated as such—and if the theory is then developed strictly deductively from an explicit set of axioms and definitions. Analogous examples taken from geometry are the Euclidean, algebraic, and projective theories. The first, as developed by Euclid, has a physical basis, while the second is mathematical, a point being defined as an ordered set of numbers (co-ordinates) and a line as the class of points satisfying a linear equation. The third is logical, the undefined elements being point and line (an undefined class of points) and the axioms being those of incidence, extension, etc. Usually a physical theory comes first, to be followed by a mathematical and then by a logical theory, this last being so constructed that it includes previous theories when its undefined elements are replaced on the one hand by the conceptual physical objects and on the other hand by the symbolic mathematical objects. The construction of such a logical theory is not merely a matter of academic interest, for it can be regarded as an analysis of the previous theories. It tests, for example, the consistency and independence of their basic assumptions and definitions. It also indicates how a theory can be modified, with as little change as possible, so as to include some feature previously excluded. This can be particularly useful in the case of a physical theory which has been constructed to correspond as closely as possible to the external world, for such a theory may need continual modification to keep in step with observational data. For this reason the axioms of a logical theory should be not only consistent and independent but also simple, i.e. indivisible.

There is a mode of specialising a quartic polynomial which causes a binary quartic to become equianharmonic and a ternary quartic to become a Klein quartic, admitting a group of 168 linear self-transformations. The six relations which must be satisfied by the coefficients of the ternary quartic were given by Coble forty years ago, but their true significance was never suspected and they have remained until now an isolated curiosity. In § 2 we give, in terms of a quadric and a Veronese surface, the geometrical interpretation of the six relations; we also give, in terms of the adjugate of a certain matrix, their algebraical interpretation. Both these interpretations make it abundantly clear that this set of relations specialising a ternary quartic has analogues for quartic polynomials in any number of variables, and point unmistakably to what these analogues are.

That a ternary quartic is, when so specialised, a Klein quartic is proved in §§ 4–6. The proof bifurcates after (5.3); one branch leads instantly to the standard form of the Klein quartic while the other leads to another form which, on applying a known test, is found also to represent a Klein quartic. One or two properties of the curve follow from this new form of its equation. In §§ 8–10 some properties of a Veronese surface are established which are related to known properties of plane quartic curves; and these considerations lead to a discussion, in § 11, of certain hexads of points associated with a Klein curve.

The quantum mechanics of fields recently developed by us leads to a modification of statistical mechanics of elementary particles which seems to overcome some of the difficulties (divergence of integrals) occurring in the usual quantum theory of fields. The main difference between the new theory and the usual one is as follows.

In the usual theory the wave-vector k is introduced classically and, so to speak, kinematically by the Fourier analysis of the field. The Fourier coefficients of the field components are then treated according to quantum mechanics as non-commuting quantities; those belonging to the wave-vector k describe the corresponding “model” mechanical system, namely the kth radiation oscillator. But the statement that the Fourier coefficients belonging to a certain k all vanish, which statement classically is significant, is now meaningless because there is a lowest state with zero-point energy for each radiation oscillator. The field is thus made to be equivalent to the assembly of radiation oscillators of all possible wave-vectors which, being necessarily infinite in number, contribute an infinite zero-point energy for the pure field and lead to other divergent integrals for the interaction between different fields.

Calculations are made of the resonance energy, bond order and bond length in a series of graphitic layers of varying size. Carbon–carbon bond lengths appear to vary very little in size with increasing number of carbon atoms, in agreement with experiment. But variations in resonance energy are significant, and indicate clearly that resonance, by itself, favours an approximately square, rather than oblong, shape. But in the case of such layers in equilibrium in the presence of molecular hydrogen, the most stable layer containing a given number of carbon atoms is of the long, thin polyphenyl type. Some tentative calculations suggest that polymerisation of smaller groups to larger ones should be endothermic, in agreement with the experimental fact that the formation of larger graphitic crystallites during carbonisation occurs, with emission of hydrogen, only at high temperatures.

The shapes of the energy bands have been determined for the mobile electrons in graphitic strips of infinite length but finite width, using as a basis the approximation of tight binding discussed for finite crystal layers in the previous paper. It appears that the effect of including overlap between the orbitals of adjacent atoms, whose incorporation in this type of calculation has hitherto been neglected, is to widen the top half of the band by a factor of the order of 2 or 3, the lower half of the band not being greatly affected. Some of these strips may be classed as conductors, the others as insulators; but the distinction between the two may not be made on the basis of any simple chemical bond diagrams.

1. 1. An n : l is defined as a group of n particles which can be contained within a seeker length l moving around a closed line of length L on which N particles are distributed at random. An expression has been obtained for the average number of distinct n : l's per closed line.

2. 2. An expression has also been derived for the average numbers of n : l's in the corresponding problem where the line of length L containing the N particles is open and not closed.

3. 3. Analogous problems in two dimensions are considered, in which the particles are arranged at random on a plane and the place of the seeker line is taken by an orientated rectangle. Exact expressions are given for the desired averages.

4. 4. The extension of the methods used to analogous problems in three dimensions is discussed. Exact expressions have not been obtained, but approximations are given which hold when n is much greater or much smaller than x, the average number found within the seeker area.

The interaction energy, or Van der Waals force, between a proton and a hydrogen atom in any one of its allowed quantum states is calculated in terms of the internuclear distance R by an expansion of the form

All the coefficients up to and including E5 are obtained in closed form. For values of R for which the expansion is valid, the coefficients are determined absolutely, no approximations being introduced.

In an earlier paper (Aitken and Silverstone, 1941) the problem of estimating from sample a parameter θ of unknown value was treated by adopting two postulates for the estimating function: (i) that it should be unbiased in the linear sense; (ii) that its sampling variance should be minimal.

I. There are several methods for obtaining transformations of hypergeometric functions of two variables.

Firstly, by transformation of the hypergeometric series. When the double series is rewritten as an infinite sum of hypergeometric functions of one variable, the known transformation theory of such functions can be applied to each term. This method is quite simple and, in a limited range, very effective for discovering transformations as well as proving them.

Secondly, by transformation of the systems of partial differential equations satisfied by the hypergeometric functions. This method, though simple in theory, is rather laborious in practice and not very useful for discovering new transformations.