The problem of the equivalence of binary forms is of great importance, both historically and intrinsically, and is also significant for the problems of the so-called canonical and automorphic forms. It consists in deciding whether two given forms are equivalent, i.e. transformable one into the other linearly, and, when they are, in finding all the linear substitutions transforming one form into the other.

Let Rc(n) be the number of representations of any non-negative integer n as the sum of c squares; i.e. Rc(n) is the number of different solutions (x1, x2, …, xc) of the equation

where the xμ are integers, and may be positive, negative, or zero. Two such solutions, (x1, x2, …, xc) and are only considered to be the same solution if

If f is a real, indefinite, binary quadratic form of discriminant d and if κ(f) is the minimum of | f | taken over all integer values of x, y, not both zero, then it is well known that and that this is a ‘best possible’ result.

It was shown by Mehler (1866) that

where Hk(x) denotes the Hermite polynomial

(Hermite, 1864a, b), which can be expressed in terms of Weber's parabolic cylinder function (Whittaker, 1903). The series is convergent if | ρ | < 1, and divergent if | ρ | > 1. If ρ = 1 and x = y = 0 the series is divergent, and Hille's work (1938) shows that it will therefore be divergent for all real or complex values, except possibly real positive values, of x and y.

A collineation in a space of paths is defined as a point transformation which carries paths into paths. Such transformations were first studied by L. P. Eisenhart and M. S. Knebelman. Subsequently J. Douglas introduced the geometry of K-spreads and E. T. Davies has shown that the results for an affine space of paths can be extended to these more general spaces and written in a very elegant form by the aid of Lie derivation.

A possible degree of heterogeneity in the structure of continua is investigated in this paper.

The Vitali covering principle is a powerful method in a wide class of problems of the theory of functions of a real variable and of the theory of sets.

During the course of some investigations on the distribution of stress in an elastic solid it was noticed by the senior author that a systematic application of the method of integral transforms to the problem of the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch reduced the problem essentially to one of solving a pair of integral equations belonging to a class which has been studied by Titchmarsh and by Busbridge (4,5). This procedure allows one to obtain the solution for an arbitrary shape of punch by a general method which leads automatically to the solution and avoids the troublesome procedure, adopted by Love (2) in the case of a conical punch, of being obliged to guess appropriate combinations of solutions which will satisfy the prescribed boundary conditions in any special case. Moreover, it can easily be seen that an attempt to apply Love's method to more complicated shapes of punch will lead to considerable analytical difficulties.

1. I. N. Sneddon has recently considered(11) the two-dimensional problem of the stress distribution due to a force uniformly distributed along a strip in the interior of a semi-infinite elastic medium by integrating the distribution due to an isolated force. The method of solution for the isolated force problem differs from that originally used by Melan(9), but both writers employ real variables. A solution of this problem, using complex variables, has been given by Stevenson and a considerable saving of labour is revealed. A complex variable method of solution for problems of stress distributions in an elastic medium bounded by a plane face was, however, given earlier by Sen(10), although he only made applications to problems in which the force system was applied to the plane face. Sen's method is somewhat different from that used by Stevenson and gives results more directly for many problems, including the problem considered by Sneddon.

1. The reflexion of waves on the surface of water by a thin plane vertical barrier is considered and the coefficient of reflexion (the ratio of the amplitudes, at a great distance from the barrier, of the reflected and incident waves) is calculated. If the top edge is at a depth a below the surface, it is found that the coefficient of reflexion is about ¼ when where T is the period of the incident waves, so that the condition that the coefficient may exceed ¼ is a .

Let γ be an irreducible variety of dimension d, and let In denote an involution of order n on Γ. The image of In is a variety C, also irreducible and of dimension d, and Γ is said to be a multiple of C. Among the possible multiples of a given variety C there is a class of particular interest; this arises when the involution In possesses the following properties:

(1) it is unramified, that is, in every set of the involution the n points are all distinct;

(2) it is generated by an Abelian group G of order n of birational transformations of Γ into itself. Given any point P of Γ the n transformations of G transform P into n points P1 = P, P2, …, Pn which constitute a set of In. The two conditions together imply that the function field κ of Γ is an unramified Abelian Galois extension of the function field k of C, and this shows that the study of the particular class of multiple varieties—which we may call normal multiple varieties when it is necessary to give a name to them—is analogous to the study of a well-known branch of the theory of numbers.

In recent years, the theory of integral transforms—often in the guise of ‘operator calculus'—has been employed to obtain the solutions of a wide variety of problems in mathematical physics, particularly in the theory of vibrations. The most commonly employed transforms are those of Fourier, Laplace and Mellin; in problems where the range of one of the variables is (0, ∞) or (− ∞, ∞) these transforms have often been used with success to reduce a partial differential equation in n independent variables to one in n − 1 variables and hence to simplify the boundary value problem involved. In this paper we consider some simple problems in the theory of elastic vibrations in which the Bessel, or Hankel, transform is more useful than those mentioned above and is consequently employed throughout. In the first instance we consider problems in which the variable to be eliminated from the differential equation ranges from 0 to ∞ so that the theory of Hankel transforms is directly applicable; the application of finite intervals is then extended to the Hankel transform by a method similar to that used by Doetsch in his treatment of the Fourier transform.

In this note I consider the Abelian integrals of the first kind on an algebraic curve Γ which is a normal multiple of a curve C, as defined in Note I*.

The problem of conduction of heat in a solid whose surface is in contact with fluid, which is so well stirred that its temperature is constant throughout its mass, often arises in practice, for example in calorimetry, or in the warming of a closed room. Despite their importance, problems of this type have not been much studied since the presence of the mass of fluid at the surface gives rise to a boundary condition for which the classical methods of solution do not apply without modification; the Laplace transformation method, however, has the advantage that it can be applied in the same way to all cases.

1. In a recent paper(1) the partial differential equation governing the symmetrical vibrations of a thin elastic plate was reduced to an ordinary differential equation by the use of the Hankel transform method. By the discussion of the solutions of the latter equation and by the use of the Hankel inversion theorem an account was given of the free and forced vibrations of the plate under symmetrical conditions.

1. In the complete algebraic system of concomitants of a pair of quaternary quadrics given by Turnbull† there appear sixteen covariant line-complexes, eight of the second order and eight of the third. In a later paper‡, the same author gave some geometrical interpretations of these and other concomitants. None the less, no systematic geometrical account of the relations of these complexes seems to have been made. The purpose of the present paper is to discuss the geometrical relations systematically, and make their origin clearer. Incidentally, the syzygies which exist between the complexes are obtained explicitly.

1. In this paper I establish in a certain sense all the relativistically invariant sets of linear partial homogeneous differential equations in which every unknown function u satisfies being the relativistic Laplace operator. Analogous questions for any orthogonal group are dealt with in a previous paper, but it was thought that an independent and not too technical treatment for the Lorentz group might be of interest.

Any particular form of mechanics (e.g. classical or quantum) makes use of a particular ‘formalism’ or set of rules governing the use of the symbols representing dynamical variables and the symbols (such as +, =) representing relations between dynamical variables. In the present paper an attempt is made to examine the physical content of this formalism, particularly that of quantum mechanics. This is done by building up a formalism as the direct expression of a number of physical postulates having direct operational meaning. It is shown that, in a simple case, with any two observables A and B can be associated a unique sum observable A + B and a unique (real) symmetric product observable A.B (= B.A).

It is next shown that, if, like a classical Hamiltonian system, the system has the property that the time rate of change of an observable depends linearly on some observable H when the environment in which the system moves is varied, then a (real) skew product observable A × B (= − B × A) can be defined.

Finally, if the equations of motion are ‘of second order', it is possible to express the second rate of change of any observable as an algebraic function of that observable and H. This leads to algebraic identities from which it follows that ‘complex multiplication’, defined by AB = A.B + iA × B, is associative (but not commutative). Observables are thus shown to possess the properties usually ascribed to them in quantum mechanics. These properties make possible a representation by Hermitian matrices.