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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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 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.

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.