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.

During the past four years I have been developing a new method of measuring the velocity of light. I had intended to defer publication until the method was perfected, but progress is becoming increasingly difficult owing to war conditions. So I give here an account of the progress already made. I am certain the method has a great future before it. There is a notice of it in a letter to Nature (Houstoun, 1938), but except for this no description of the method has yet been given.

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.

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.

A quasi-field is denned by the postulates of a commutative algebraic field, except that the distributive law a(b + c) = ab + ac is replaced by a(b1+ … +bn)=ab1+ … +abn for a fixed integer n.

The properties of quasi-fields are investigated. The study of their ideals is reduced to the study of the ideals of a certain type of ring. A particular quasi-field is constructed formally by means of polynomial domains modulo a natural number, with addition specially defined.

Quasi-fields are connected with multiple fields—another generalisation of the conception of a commutative field, in which a fixed number of elements (> 2) co-operate symmetrically in the formation of any sum or product.

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.

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.

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 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 this paper the equations of the meson are treated in the same manner as the Dirac equation in a previous paper (K. Fuchs, 1940, Part IV). We use the formulae developed by Kemmer (1939), with a small modification, introduced so that the set of Maxwell's equations for the electro-magnetic field is obtained as a special case.