Under the condition that one at least of the leading coefficients amn, a0n differs from zero, the equation

has as solution a series convergent for all x greater (or all x less) than a fixed number. The coefficients of the various terms in the series are expressed in terms of the arbitrary values of the solution and its first n derivatives in an initial interval of appropriate length.

This paper was assisted in publication by a grant from the Carnegie Trust for the Universities of Scotland.

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.

In many problems in physics and chemistry certain determinants of large order and of a special type require to be evaluated. The determinants in question usually arise from some kind of secular equation, and in most cases they owe their origin to a system of particles of one kind or another which, in their equilibrium positions, form a regular lattice, each particle being acted upon by its nearest neighbours and perhaps by a fixed boundary. These forces may be assumed to be elastic in character, to the degree of approximation required. For simplicity we shall refer to these particles as atoms although they may be electrons, molecules or even material particles in the Newtonian sense, according to the nature of the problem under investigation. In particular we may mention the occurrence of these determinants in problems involving the solution of Schrödinger's Wave Equation for permitted energy levels and in determining the distribution of electric charge in crystals, metals and large molecules. They have also been used by Born in his investigations of crystal structure by means of X-rays. It is hoped therefore that the results achieved in this paper will be of interest to the physicist and chemist as well as to the mathematician.

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.

A set of variables is assumed to depend upon a number of common factors and specifics. Formulae are then derived for the sampling variances and covariances of the residual covariances obtained by removing the effect of the factors. The variances and covariances of the set of estimated loadings are also found. It must, however, be noted that the results obtained are valid only when an efficient method of estimation is used.

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.

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.

I. Introduction.—Consider a Hamiltonian system of differential equations

where H is a function of the 2n variables qi and pi involving in general also the time t. For each given Hamiltonian function H the system (1.1) possesses infinitely many absolute and relative integral invariants of every order r = 1,…, 2n, which can all be written out when (1.1) is integrated. Our interest now is not in these integral invariants, which are possessed by one Hamiltonian system, but in those which are possessed by all Hamiltonian systems. Such an integral invariant, which is independent of the Hamiltonian H, is said to be universal.

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.

28. This paper contains the investigation of certain properties of periodic solutions of Lamé's differential equation by means of representation of these solutions by (in general infinite) series of associated Legendre functions. Terminating series of associated Legendre functions representing Lamé polynomials have been used by E. Heine and G. H. Darwin. The latter used them also for numerical computation of Lamé polynomials. It appears that infinite series of Legendre functions representing transcendental Lamé functions have not been discussed previously. In two respects these series seem to be superior to the generally used power-series and Fourier-Jacobi series, (i) They are convergent in some parts of the complex plane of the variable where both power-series and Fourier-Jacobi series diverge, (ii) They permit by simply replacing Legendre functions of first kind by those of second kind, to deal with Lamé functions of second kind as well as Lamé functions of first kind (§ 15).

In §§ 2 and 8 of the present paper the series are heuristically deduced from the integral equations satisfied by periodic Lamé functions. Inserting the series found heuristically, with unknown coefficients, into Lamé's differential equation, recurrence relations for the coefficients are obtained (§§ 9–12). These recurrence relations yield the (in general transcendental) equations in form of (in general infinite) continued fractions for the determination of the characteristic numbers. The convergence of the series can be discussed completely.

There are altogether forty-eight different series. Every one of the eight types of Lamé polynomials is represented by six different series (see table in § 13). There are interesting relations (§ 14) between series representing the same function.

Next infinite series representing transcendental Lamé functions are discussed. It is seen that transcendental Lamé functions are only simply-periodic (§§ 18, 19). Lamé functions of real (§§ 20–22) and imaginary (§§ 23-24) period are represented by series of Legendre functions the variables of which are different in both cases.

The paper concludes with a brief discussion of the most important limiting cases, and a short mention of other types of series of Legendre functions representing Lamé functions.

The title I have chosen for this lecture contains an implication which perhaps will not be generally accepted; there was a time when I would not have accepted it myself. The implication is that a particular kind of philosophy is possible which may be called scientific in contrast with other kinds which cannot be so called. I would go further and identify this scientific philosophy with what is generally called science, and this implies that the distinction that is often assumed to exist between science and philosophy is a false one. For this view I believe there is historical evidence. Science, as a separate, self-contained study, dates from the seventeenth century. Before that time, such consideration as was given to the subject-matter of present-day science was given it by philosophers and regarded as a part of their philosophising, and when in the seventeenth century a new kind of procedure was introduced, it was looked upon by its pioneers not as an attack on a new problem but as a new attack on an old problem. The science of that time was the “new philosophy”, faintly adumbrated by some mediæval philosophers, struggling for expression in Francis Bacon, and coming to full recognition in Galileo. Only later, when it had made such progress in certain limited fields of study that a new body of investigators was called into being who confined themselves to those fields, was the new philosophy transformed into a non-philosophy and called generally by the name “science”.

R. Frisch, in a paper (Frisch, 1928) on correlation and scatter in statistical variables, made an extensive use of matrices, and in particular of the moment matrix, as he called it, of a set of variables. The matrices were square arrays, with an equal number of rows and columns. This paper of Frisch pointed the way to an even more extensive use of the algebra of matrices in problems of statistics.

What Frisch called the moment matrix may perhaps be more suitably called, nowadays, the variance matrix of a set or vector of variates, since the moments in question are all variances or covariances. In the present paper, which is illustrative of matrix methods, we explore the familiar ground of linear approximation by Least Squares, making full use of the properties of the variance matrix. We also study the linear transformations that convert crude data into smoothed or graduated values, or into residuals, or into coefficients in a linear representation by chosen functions.

I. By the discriminant D of a homogeneous polynomial ø is, in accordance with the general custom, to be understood that function of its coefficients whose vanishing is the necessary and sufficient condition for the locus ø = o to have a node. It is the resultant, or eliminant, of the set of equations obtained by equating all the first partial derivatives of ø simultaneously to zero. If ø contains n variables and is of order p, the degree of D in the coefficients of ø is n(p–I)n−1.

In classical mechanics the mass of a system of gravitating particles can be denned to be the mass of an equivalent particle which gives the same field at great distances, or alternatively the mass can be defined by means of Gauss' Theorem. Reference to the former procedure was made by Eddington and Clark (1938) in a discussion on the problem of n bodies. The relativistic extension of Gauss' Theorem has been investigated by Whittaker (1935) for a particular form of the line-element and for more general fields by Ruse (1935). The latter, treating the problem from a purely geometrical point of view, expressed the integral of the normal component of the gravitational force as the sum of two volume integrals. The physical significance of one of these integrals was quite obvious but the meaning of the other was far from clear. In this paper the terms in Ruse's result are examined as far as the order m2 in the case of a fundamental observer at rest and the 1938 discussion modified to bring the two investigations into line. It is concluded that the surface integral of the normal component of the gravitational force taken over an infinite sphere is –4π × the energy of the system.

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 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.

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.

If is a fixed point of a Riemannian Vn of fundamental tensor gij, and if s is the geodesic distance between it and a variable point (xi), then the Vn has been called centrally harmonic with respect to the base-point if

is a function of s only, and completely harmonic if this holds for every choice of base-point . A flat Vn (gij=δij) is obviously completely harmonic, since for such a space and

Throughout the preceding discussion we have considered only those contributions to the stresses which are due to either the motion of the body or the presence of other bodies. That is, we have not considered the stresses which occur in the systems discussed by Whittaker, and we have, in effect, assumed that the gravitational mass of an isolated body at rest is the same as its invariant mass at least as terms of order m2 are concerned. In this appendix we complete the investigation by demonstrating the validity of this assumption.