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.

An account is given of Professor Marcel Riesz's generalisation of the Riemann-Liouville integral of fractional order. It is shown that the new ideas introduced by Riesz may prove valuable in the theory of partial differential equations and in the theory of the wave-equation in momentum space.

9. A formula is adopted which gives the probability of an individual of given ability passing a test item in terms of two quantities constant for that item. A method of estimating these two constants is given. Making certain assumptions concerning the items composing a test, formula? are then derived giving the expected value and the standard error of the test score of any person. It is shown that there is a certain reciprocity between persons and items, and corresponding formula? are given for the item scores. Finally, the results are applied to actual data obtained on a Moray House intelligence test, an estimate being made of the reliability of the test.

In conclusion I should like to thank Professor Godfrey H. Thomson for his help and valuable criticism in connection with this paper. I should also like to take this opportunity of thanking the Carnegie Trust for the Universities of Scotland for grants to cover the cost of the setting and printing of mathematical formula? in two papers previously published in the Society's Proceedings.

In a recent paper in these Proceedings, Dr G. C. McVittie has published some criticisms of kinematical relativity. These criticisms are to a large extent based on his formula (4.10), namely,

It must be stated at the outset that McVittie's interpretation of his derivation of (1) as a derivation of “Milne's formula for the acceleration of a ‘free particle moving in the presence of a substratum,’ for the special case of one spatial co-ordinate only” is wrong. McVittie does not derive the result, as he claims, from what he calls the “axioms of kinematical relativity” alone; he deduces it from these axioms together with an additional assumption, which is equivalent to begging the answer to the whole problem it was my object to solve. Instead of considering a free particle, as I did—that is, a particle whose motion we do not a priori know—he prescribes a priori the motion of his particle as being constrained to obey the rule, in his notation,

In the system of two linear partial differential equations of the second order

a,…,f were supposed to be polynomials in x, and a1…, f1 polynomials in y. These polynomial coefficients were subjected to certain restrictions, including conditions for the system having exactly four linearly independent solutions, and conditions for preserving the symmetrical aspect, in x and y, of the system. It has been proved that any compatible system of the contemplated form whose coefficients satisfy the stipulated conditions is equivalent with, i.e. transformable into, a hypergeometric system. More particularly it has been shown that the hypergeometric systems involved are the system of partial differential equations associated with Appell's hypergeometric function in two variables F2 and the confluent systems arising herefrom.

The suggestion has recently been put forward that the laws of nature can be established by purely deductive reasoning instead of by induction from observation. We may, with Eddington, start the chain of reasoning from epistemological premises or, with E. A. Milne, from axiomatic statements regarding the nature of the system to be studied. Different opinions may be held regarding the value of a deductive method, but a final judgment can hardly be passed on a deductive theory until the initial premises are clearly revealed. We may, indeed, justly require of the author of such a theory that he fulfil the following conditions. He should, firstly, be himself aware of all the axioms which he employs. If he is not, there is the obvious danger that he may use inductions from observation without being aware of doing so. But he may also arrive at quite erroneous conclusions about the range of validity of his results. For instance, a deductive theory may produce a formula which is interpreted as the inverse square law of gravitation. It is then very necessary to know whether the initial premises are axioms concerning the nature of the universe as a whole or whether they merely define local conditions. In the first case the law of gravitation is deduced from the nature of the universe as a whole, in the second it is shown to be merely a “local” law.

The object of this paper is to give some numerical results for the cooling of the region bounded internally by a circular cylinder, with constant initial temperature, and various boundary conditions at the surface. Problems of this nature are of importance in connection with the cooling of mines, and in various physical questions.

The case of constant surface temperature is discussed in § 2. In § 3 the results are compared with the corresponding ones for the region outside a sphere, and for the semi-infinite solid.

In his recent article McVittie has criticised in most disparaging terms the analytical theory of time-keeping developed by Milne and myself in the last ten years. Milne has replied at length (see preceding paper), and it is my purpose in this note merely to touch on one or two points which he has not covered.

However, before doing so, I should like to take this opportunity of remarking that the “Kinematical Relativity,” about which McVittie has written, both in his recent article and in his monograph, “Cosmological Theory,” is not the Kinematical Relativity of Milne and myself, but is something much slighter, based, perhaps, on an incomplete understanding of the nature of the kinematical theory.

The paper makes use, for the study of a ternary quartic, of a five-dimensional configuration consisting of a Veronese surface and a quadric outpolar to it, and uses the notation and results of a preceding paper to which reference is made at the outset. In § 1 certain identities are given which are consequences of the form of the matrix of a quadric outpolar to a Veronese surface, and the geometrical theorems equivalent to these identities are stated. In § 2 it is explained how covariants and contravariants of a ternary quartic are represented by curves in the fivedimensional configuration. It is, indeed, not until this technique is used that some of the work of Clebsch, Ciani, Coble, and others is properly appreciated; §§ 3—6 are concerned to emphasise this. But, as is pointed out in § 7–9, it is to Sylvester that these matters must properly be referred; for he has, by his process of unravelment, anticipated practically everything of moment in the ideas of his successors. The word unravelment is used by him on p. 322 of Vol. I of his Mathematical Papers, the process having appeared on p. 294.

In opening the second part of the paper with § 10 it is pointed out that the configuration should be used not merely to illuminate the work of previous writers, but also to discover new results. It is not the purpose here to exploit this at length, but it is seen how a covariant conic inevitably appears; its equation is obtained and, in §11, its covariance directly established. Other covariant conies are alluded to in § 12. And it is found, in § 13, that here too reference must be made to Sylvester.

IT falls to us this year to commemorate the greatest of men of science, Isaac Newton, on the occasion of the three-hundredth anniversary of his birth. The centuries have not dimmed his fame, and the passage of time is unlikely ever to displace him from the supreme position. His discoveries, however—and this is part of their glory—have not persisted unchanged, but in the hands of his successors have been continually unfolding into fresh evolutions. During the eighteenth and nineteenth centuries there was an immense expansion of knowledge, springing directly from his work, and forming ultimately a vast superstructure based on the Newtonian concepts of space, mass, and force. Since 1900 the progress of science has continued, but the development of physics has changed in character: it has become subversive and radical, questioning the traditional assumptions and uprooting the old foundations. In 1915 the Newtonian doctrine of gravitation was superseded by that of Einstein: the divergence between the results of the two theories, so far as concerns the calculation of the movements of the planets, is extremely slight, and indeed, in almost all cases, too small to be detected by observation; but on the question of the essential nature of gravitation, the two conceptions differ completely and are associated with opposite philosophies of the external world. The other great discovery of the present century is the quantum theory, which in its perfected form of quantum-mechanics appeared in 1925: this also is completely irreconcilable with the postulates of Newtonian science.