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 The unpublished work mentioned above is not required for this purpose because the cosmological models considered here were described in earlier papers (Walker, 1937, 1940 b).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.

Substitutional equations of the type considered by the late Alfred Young are shown to be intimately related with the theory of idempotents. Any equation LX = o possessing solutions other than X = o is shown to have the same solutions as another equation MX = o, where M is obtained from L by premultiplying the latter by a suitably chosen expression A and where the minimum equation of M is xψ(x) = o, ψ(x) being prime to x. The expression ψ(M) is then idempotent, and it is shown that the most general solution of LX = o is X = ψ(M)Y, where Y is an arbitrary expression. The number of linearly independent solutions of LX = o is X = ψ(M)Y, where Y is an arbitrary expression. The number of linearly independent solutions of LX = o is kn!, where k is the coefficient of the unit permutation in ψ(M) when that expression is expressed in terms of the permutations of the symmetric group Sn.

Corresponding results are obtained for the equation LX = R, and methods are given for solving sets of simultaneous equations of both types.

Studying the interaction of different pure fields, we have been led to some essential modifications of the ideas on which our quantum mechanics of fields is based. We shall explain these here for the example of the interaction of the Maxwell and the Dirac field.

In Part I we showed that a pure field in a given volume Ω can be described by considering the potentials and field components as matrices, not attached to single points in Ω (as the theory of Heisenberg and Pauli), but to the whole volume. Further, we assumed the total energy and momentum to be the product of Ω and the corresponding densities. In Part † we showed that this conception has to be modified; the eigenvalues of the energy and momentum as defined in Part I represent neither the states of single particles nor of a system of particles, but of something intermediate which corresponds to the simple oscillators of Heisenberg-Pauli and which we have called apeirons. The total energy and momentum of the system is a sum over the contributions of an assembly of apeirons. Mathematically the differences of the quantum mechanics of a field from that of a set of mass points (as treated in ordinary quantum mechanics) is the fact that the matrices representing a field are reducible (while those representing co-ordinates of mass points are irreducible); each irreducible submatrix corresponds to an apeiron.

The changes in his description of events brought about by an arbitrary regraduation of an observer's clock are examined, taking the axioms of general relativity as fundamental. It is shown that regraduation does not imply a change from one Riemannian space-time to another but merely a coordinate transformation within space-time. A generalisation of the “dynamical time” of kinematical relativity is a by-product of the investigation.

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.

6. The expected score of an individual on a test consisting of a large number of items is assumed to be given by a formula involving the ability of the individual and also two quantities constant for the test. An expression is then derived for the covariance between two tests measuring different abilities. It appears that if a factorial analysis is performed on a set of tests of unequal difficulty, using the matrix of variances and covariances, a spurious factor will tend to be introduced depending mainly on the differences in difficulty. The effect of this is removed by transforming the variances and covariances to a new set of coefficients. A numerical example of the process is given.

In conclusion I should like to thank the Carnegie Trust for the Universities of Scotland for a grant to cover the cost of the setting and printing of mathematical formulæ in a paper previously published in the Society's Proceedings (LXI, A, 1943, 273–287).

In the Proc. Roy. Soc. Edin. (Houstoun, 1941) I described a new method of measuring the velocity of light. Owing to the difficulties of the times progress has been slow, but I have now succeeded in measuring the velocity of light in water and communicate the result here.

Previous Measurements.—Every elementary textbook on Light explains how Newton's, corpuscular theory required the velocity of light in water to be μ times the velocity in air, whereas Huygens's wave theory required it to be I/μ times the velocity in air, and how Foucault's determination decided between the two theories. As a matter of fact it did, but Foucault's determination was not an accurate one and there have been only four previous determinations of the velocity of light in water, none of which can lay claim to accuracy. These are the determinations of Foucault (1850), Fizeau and Bréguet (1850), A. A. Michelson (1891), and Gutton (1911).

Wave functions have been obtained for the ground states of the atoms Li to Ne. These are of the simple analytical type proposed by Morse, Young, and Haurwitz. Some errors in the latter work are corrected.

In 1902, Professor E. T. Whittaker gave a general solution of Laplace's equation in the form

where f is an arbitrary function of the two variables. It appears that this is not the most general solution, since there are harmonic functions, such as r−1Q0(cos θ), which cannot be expressed in this form near the origin. The difficulty is naturally connected with the location of the singular points of the harmonic function. It seems therefore to be worth while considering afresh the conditions under which Whittaker's solution is valid.

In this paper the curvature tensor Rijkl in a Riemannian Vn is used to define a quadratic complex of lines in an (n – I)-dimensional projective space Sn–1. Work in this direction has been done for a V4 by Struik (1927–28), Lamson (1930), and Churchill (1932). Of these, Struik and Lamson both use 3-dimensional projective geometry, the former for the purpose of defining sets of “principal directions” in V4 by means of the Riemann tensor, and the latter for the purpose of discussing some of the differential and algebraic consequences of the field equations of general relativity. Churchill considers the geometry of the Riemann tensor from the point of view of 4-dimensional Euclidean geometry. In this paper an indication is given of the nature of the general n-dimensional theory, which, by way of elementary illustration, is then applied in moderate detail to a V3. A few general formulae are also obtained for a V4.

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 difficulties met in the usual treatment of quantised field theories seem to us somewhat similar to those which occurred in Bohr's semi-classical quantum mechanics of particles. In this theory the orbits were described by Fourier series in the time; there was no exact correspondence between the periodic terms of this series and quantum transitions, but only an approximate one for terms of high order. Matrix mechanics considers not the Fourier series, but the single terms which are generalised into matrix elements having not one but two indices. This generalisation is founded on Ritz's combination principle.

It is a well-known fact that Karl Pearson's formulae expressing the effect of selection on the means, variances and covariances of a multivariate population hold when the variates are such as to be normally distributed both before and after selection. It is not, however, generally known that the formulae are true under much more general conditions, and in view of a recent controversy it has been thought desirable to establish precisely what these conditions are. In dealing with the problem we shall adopt the shortened vector and matrix notation introduced by Aitken (I). This notation is reproduced below with but slight modifications.

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.

1. In a recent paper Professor G. Temple has given a matrix representation of the Clebsch-Aronhold symbols by means of which a homogeneous form of degree m in the n variables x1 …, xn may be written (a1x1 + … + anxn)m. The present paper is concerned with an extension of Temple's method to include Weitzenböck's complex symbols which have proved so potent in the treatment of linear and higher complexes. A slight rearrangement of Temple's matrices is suggested which displays more clearly the nature of the representation.

The term synthetic plastic covers such a wide variety of substances that it is an extremely difficult matter to attempt a definition—nor is it necessary for the purpose of this review. To the chemist plastics are solids of high molecular weight exhibiting properties which lie between those of liquids and solids: at sufficiently high temperatures they behave like liquids and at low temperatures like solids. These substances do not obey many of the laws which form the corner-stones of chemical theory, for their chemical constitution can be varied by indefinitely small degrees and it is possible mechanically to divide the molecule into parts. To the layman, on the other hand, plastics have come to mean substances used to fabricate, by mass-production methods, relatively trivial articles like ashtrays, ornamental door-knobs, etc. They tend to be regarded as substitutes which must necessarily be inferior to the materials, once used to make these domestic articles. Much, too, has been written to give the impression that we are on the verge of the plastics era. This is an optimistic exaggeration. Plastics are designed to play an important part as unique materials which can be built by the chemist to the specification of the engineer and the physicist. It is wrong to regard them as substitutes. They are new materials and must be used as such. The purpose of this survey is to indicate how synthetic plastics fit into a future economy in which their special chemical and mechanical properties find their proper application.

From the time of Galileo, experiment has been the core of Natural Science. Before him, of course, observation alone had in the development of astronomy played a fundamental part. Besides the great workers of the ancient civilisations, who knew the path of the sun amongst the fixed stars and could predict eclipses, and besides the fruits of Greek astronomy associated with the names of Hipparchus and Ptolemy, the more modern observational work of Tycho Brahe, analysed by Kepler, had vindicated the self-consistency of the Copernican theory of the solar system and had led to its remarkable refinement in the form of Kepler's three quantitative laws—the law of the ellipse, the law of areas, and the law connecting periodic times and major axes. This was a triumphant example of the execution of the programme then being put forward by Francis Bacon for discovering all natural laws—the method of induction from a number of instances. But it was reserved for Galileo to make a start with the process of ascertaining as far as might be, by controlled experiment, the particular nature of motion. The metaphysical questions associated with motion had not escaped the attention of the Greeks; but Zeno was apparently content with stating paradoxes, and did not resolve them. Galileo, first, experimented with moving bodies; and established that in falling they received equal increments of velocity in equal times—a kinematic theorem, like Kepler's laws. Huyghens was perhaps the first person to establish dynamical-theorems; that is to say, to infer a kinematic result from a stated physical principle—as, for example, his proof of the approximate isochromism of the pendulum based on the principle of vis viva, or, as we should now say, the conservation of energy. Huyghens, together with some of the early Restoration men of science in this country, dealt also with the collisions of bodies. The peerless Newton went further. Assuming outright three primitive “laws of motion,” he showed how the results of Galileo, Huyghens, and their contemporaries could be actually deduced; and by the addition of a fourth law, the law of universal gravitation, already conjectured by some thinkers, he arrived at the laws of Kepler as inferences. Not only so, but the four highly general and abstract laws introduced by Newton have been found sufficient to deduce an enormous complex of dynamical theorems, to express their relationships in the subsequent beautiful systems of Lagrange and of Hamilton, and to derive all but every detail in the motions both in the solar system and in distant binary stars. The basic principles laid down by Newton remained unaltered till our own day, when Einstein modified simultaneously the laws of motion, the law of gravitation, and the background of space and time which had been explicitly adopted by Newton as the scene in which his laws were to play their parts.