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.

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.

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.

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.

The problem with which this paper is concerned arose in the discussion of a series of chronometric observations, but it is of more general application, and is capable of wide extension. Pairs of readings (xi yi) were taken at times ti, i = 1, 2, …, n. These readings were known to be affected by respective errors (ξi ηi) from sources different but possessing some common part. It was important to have an estimate of the consequent correlation and to assess its precision. The assumptions made in the particular experiment were that x and y were both linear in t, representable by x = a0 + a1t, y = b0 + b1t, and that the distributions of error in x and y were normal. The parameters a0 and a1, b0 and b1 were therefore obtained from two separate sets of normal equations, and the unknown correlation was then estimated from the sum of products of corresponding residuals ui, vi, one from each set. In the corresponding situation in n samples (xi,yi) from a bivariate normal distribution the mean value of is (n − 1) ρσ1σ2, where σ12, σ22 are the variances of x and y and ρσ1σ2 is their product moment. One might therefore anticipate, by analogy, that in the present case the mean value of Σuivi would be (n − 2)ρσ1σ2. So indeed it proves to be, and the sampling variance of Σuivi conforms likewise with standard results; but it is desirable, by an extension of the problem, both to see why this is so and to take notice of cases where the analogy fails to hold.

Hill's differential equation (1.1) derives its importance from being the prototype of the different equations of Lamé and of the equation of Mathieu, which are connected with wave and potential problems in mathematical physics. Besides this, numerous instances of its occurrence in problems of elasticity and of dynamical or statical stability are known. In the present treatment, conditions are reversed with respect to most of the older publications, since the characteristic multiplier σ of equations (1.2) is not sought as a function of the given parameters λ and γ of equation (1.1), but σ is supposed given and the corresponding values of λ and γ are regarded as unknown. Thus a linear homogeneous boundary value problem of the second order and of non-self-adjoint type ensues, the values of σ and of λ, σ being in general complex. On this latter point the present paper considerably enlarges the scope of some previous papers published by the author during the war along somewhat similar lines but for real characteristic values (Nos. 13–18 of the references at the end).

Thermal diffusion in aqueous solutions of raffinose, sucrose, glucose, xylose, glycerol and acetone was studied in respect of variation of concentration and of temperature. Initial rates of separation were determined as produced by an apparatus consisting of two vertical concentric glass tubes of length 130 cm. and mean annular separation o·68 mm.

With a solution of glucose of 1·7 gm. mol. per litre, the initial rate of separation for the extremes of temperature which could be experimentally applied was found to be independent of the column length for lengths greater than 100 cm.

With the exception of acetone, all the solutes concentrated at the column foot. In singlesolute solutions a maximum rate was observed at an intermediate concentration, which in the sugar series was at a lower molar concentration the higher the molar weight of the solute. With the two-solute solutions sucrose-glucose and sucrose-glycerol little or no relative separation was obtained, contrary to expectation based on the single-solute data.

Initial rates increased with rise in mean temperature and with increase in the applied temperature gradient.

An elementary theory of thermal diffusion applicable to gaseous and liquid systems has been developed. This is based on the difference of diffusional characteristics of a molecule considered as diffusing through two different temperature regions, when the pressure is constant throughout.

For gaseous systems, the resultant expression is shown to be in general accordance with experimental variation of temperature, mass, and diameter factors, and is further developed to include isotopic separation, change of sign of separation with concentration, and general force law considerations.

A similar approach to thermal diffusion in solution, combined with the convection effect of a “cascade” system, gives an expression which is in general agreement with the results of experimental variation of mean temperature and temperature gradient for aqueous solutions of sucrose, glucose and glycerol. The simple expression does not account rigidly for the sign of separation or the effect of altered concentrations. These discrepancies are discussed in relation to the general formula; it is concluded that in addition to the diffusion diameters, the relative thermal expansions of solute and solvent are of importance in this connection.

A general formula is obtained for the interference velocity when an aerofoil with elliptically distributed circulation is in a closed or open wind tunnel of any cross-section. The mapping of the section on the interior of a circle is given in terms of the Jacobian elliptic functions appropriate to the ellipse and rectangle. The result is worked out for an aerofoil which spans the focal distance in a tunnel whose section is an ellipse.

Relativity is the study of matter in motion, and the basis of a theory of relativity can be either physical, mathematical, or logical. It is physical if some of the elementary objects and relations are concepts derived from the external world and if certain of their properties are assumed as physically obvious. If, however, the elementary objects, etc. are defined as mathematical symbols and relations, and if the subsequent theorems are mathematical deductions from these definitions, then the theory may be described as mathematical. Lastly, the basis of a theory is logical if certain terms are undefined—and clearly stated as such—and if the theory is then developed strictly deductively from an explicit set of axioms and definitions. Analogous examples taken from geometry are the Euclidean, algebraic, and projective theories. The first, as developed by Euclid, has a physical basis, while the second is mathematical, a point being defined as an ordered set of numbers (co-ordinates) and a line as the class of points satisfying a linear equation. The third is logical, the undefined elements being point and line (an undefined class of points) and the axioms being those of incidence, extension, etc. Usually a physical theory comes first, to be followed by a mathematical and then by a logical theory, this last being so constructed that it includes previous theories when its undefined elements are replaced on the one hand by the conceptual physical objects and on the other hand by the symbolic mathematical objects. The construction of such a logical theory is not merely a matter of academic interest, for it can be regarded as an analysis of the previous theories. It tests, for example, the consistency and independence of their basic assumptions and definitions. It also indicates how a theory can be modified, with as little change as possible, so as to include some feature previously excluded. This can be particularly useful in the case of a physical theory which has been constructed to correspond as closely as possible to the external world, for such a theory may need continual modification to keep in step with observational data. For this reason the axioms of a logical theory should be not only consistent and independent but also simple, i.e. indivisible.

Calculations are made of the resonance energy, bond order and bond length in a series of graphitic layers of varying size. Carbon–carbon bond lengths appear to vary very little in size with increasing number of carbon atoms, in agreement with experiment. But variations in resonance energy are significant, and indicate clearly that resonance, by itself, favours an approximately square, rather than oblong, shape. But in the case of such layers in equilibrium in the presence of molecular hydrogen, the most stable layer containing a given number of carbon atoms is of the long, thin polyphenyl type. Some tentative calculations suggest that polymerisation of smaller groups to larger ones should be endothermic, in agreement with the experimental fact that the formation of larger graphitic crystallites during carbonisation occurs, with emission of hydrogen, only at high temperatures.

The shapes of the energy bands have been determined for the mobile electrons in graphitic strips of infinite length but finite width, using as a basis the approximation of tight binding discussed for finite crystal layers in the previous paper. It appears that the effect of including overlap between the orbitals of adjacent atoms, whose incorporation in this type of calculation has hitherto been neglected, is to widen the top half of the band by a factor of the order of 2 or 3, the lower half of the band not being greatly affected. Some of these strips may be classed as conductors, the others as insulators; but the distinction between the two may not be made on the basis of any simple chemical bond diagrams.

The interaction energy, or Van der Waals force, between a proton and a hydrogen atom in any one of its allowed quantum states is calculated in terms of the internuclear distance R by an expansion of the form

All the coefficients up to and including E5 are obtained in closed form. For values of R for which the expansion is valid, the coefficients are determined absolutely, no approximations being introduced.

In an earlier paper (Aitken and Silverstone, 1941) the problem of estimating from sample a parameter θ of unknown value was treated by adopting two postulates for the estimating function: (i) that it should be unbiased in the linear sense; (ii) that its sampling variance should be minimal.

I. There are several methods for obtaining transformations of hypergeometric functions of two variables.

Firstly, by transformation of the hypergeometric series. When the double series is rewritten as an infinite sum of hypergeometric functions of one variable, the known transformation theory of such functions can be applied to each term. This method is quite simple and, in a limited range, very effective for discovering transformations as well as proving them.

Secondly, by transformation of the systems of partial differential equations satisfied by the hypergeometric functions. This method, though simple in theory, is rather laborious in practice and not very useful for discovering new transformations.