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.

The quotient of alternants

the denominator being the difference-product of the arguments a, β, γ, …, is an important symmetric function, introduced into algebra by Jacobi in 1841.

In the problem of estimating from sample the value of a parameter in a probability function new postulates are suggested of unbiased linear estimate and minimum sampling variance. A comparison is made, with illustrative examples, between this method and the principle of maximum likelihood, and ground common to the two is traversed. The new postulates are also placed in relation to the theory of sufficient statistics.

More than two thousand years ago the Greek philosophers raised certain questions, which are still undecided, about the origin and character of knowledge regarding the external world. After a period of comparative quiet, the discussion has become very active recently, under the stimulus of the new discoveries in mathematical physics; and, in particular, a lively debate is in progress at the present moment between Sir Arthur Eddington and Dr Harold Jeffreys of Cambridge, Professor Milne of Oxford, Sir James Jeans, and Professor Dingle of the Imperial College, the subject being the respective shares of reason and observation in the discovery of the laws of nature. I propose this afternoon to offer some remarks on the history and present state of this controversy.

The subject-matter of these pages may be briefly summarised as follows: the geometry of the Veronese surface, with an algebraic representation of it that does justice to its self-dual character; the relations of the secant planes of the surface to quadrics which either contain the surface or are outpolar to it; and the derivation of an invariant and two contravariants of a ternary quartic in the light of the (1, 1) correspondence between the quartic curves in a plane and the quadrics outpolar to a Veronese surface. There is no suggestion of discovering fresh properties of the surface, though possibly the results in § 12 § 13 may be new; but the geometrical considerations lead naturally to some algebraical results which it seems worth while to have on record, such as, for example, the identity 8.2 and the remarks concerning the rank of the determinant which appears there, and the form found in § 13 for the harmonic envelope of a plane quartic curve. These algebraical results lie very close to properties of the surface; so close in fact that one might say that the Veronese surface is the proper mise en scène for them.

Wave mechanics is able to describe with some precision the motions of electrons in atoms, but when we study molecules we have to use more approximate descriptions. It turns out that what the chemist is accustomed to call a single bond is in reality a pair of electrons, having opposed spins, describing equivalent orbits which have symmetry about the line joining the two nuclei concerned; this may be called a localised bond. The tetrahedral character of the bonds from saturated Carbon atoms are easily fitted into this scheme.

In Ethylene, however, another type of orbit appears; this is the double-streamer orbit, and two electrons in this orbit convert a normal single bond into a double bond. Again the bond is a localised bond, with a characteristic energy and length.

In more complex molecules, such as Benzene, there is a framework of single bonds, and the remaining electrons have orbits that embrace all six of the Carbon atoms; these mobile electrons give the aromatic and conjugated molecules their characteristic properties, but as a result the bonds are neither pure single bonds nor pure double bonds, but a hybrid of the two, and the electrons in these bonds are no longer localised in the region between any two particular nuclei. The energies of these molecules can be calculated in fair agreement with experiment, and from a knowledge of the wave function it is possible to define an order, which is usually fractional, for these bonds. In Benzene all the C-C links are equivalent, and their order is I⅔.

A curve which connects the fractional order with the length of the bond enables us to predict the lengths of these bonds, and, where experimental comparison is available, agreement is found. These mobile electrons are important in a study of vibration frequencies, restricted rotation about C-C bonds, and in polymerisation.

7. A method of factor estimation is given in which assumptions are made only about the form of the error distributions of the tests administered. This is compared with a method previously suggested in which, on the contrary, the test scores and the individual factor measurements were assumed to be normally distributed over the population of individuals tested. A comparison is also made with other processes at present in use.

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.