1. In this paper I shall deal with the solutions of the Lamé equation

when n and h are arbitrary complex or real parameters and k is any number in the complex plane cut along the real axis from 1 to ∞ and from −1 to −∞. Since the coefficients of (1) are periodic functions of am(x, k), we conclude ](5), § 19·4] that there is a solution of (1), y0(x), which has a trigonometric expansion of the form

where θ is a certain constant, the characteristic exponent, which depends on h, k and n. Unless θ is an integer, y0(x) and y0(−x) are two distinct solutions of the Lamé equation.

It is easy to obtain the system of recurrence relations

for the coefficients cr. θ is determined, mod 1, by the condition that this system of recurrence relations should have a solution {cr} for which

k′ being the principal value of (1−k2)½

This paper is an attempt to apply a simple series solution of the wave equation to the reflexion of a sound pulse. The reflector is a paraboloid of revolution, and the incident pulse is spherically symmetrical and comes from the focus of the reflector, so that the wave fronts of the reflected pulse are planes at right angles to the axis of symmetry of the reflector. If the incident pulse consists, at any point, of a discontinuous pressure rise followed by constant excess pressure, the series reduces to a power series in the time counted from the onset of the reflected pulse; for other forms of the incident pulse it can be interpreted as the result of the superposition of elementary pressure pulses which are constant in a small time interval and vanish outside it. No convergence proof is given, so that the interest of the investigation is physical rather than mathematical; but the numerical results indicate that the convergence of the series is unsatisfactory except near the vertex of the reflecting paraboloid. The coefficients of the series are obtained with the aid of recurrence formulae, and the first seven coefficients have been calculated. The calculations become more laborious at each successive stage. A detailed numerical investigation of the reflexion of the ‘simple rectangular pulse’ referred to already (for which the series reduces to a power series) reveals that initially the maximum pressure on any plane section at right angles to the axis of the paraboloid, due to the reflected pulse alone, occurs at the axis and has the same value everywhere on it; but after some time a secondary pressure maximum is established over a circular ring at some distance from the axis. A consideration of the initial pressure gradient of the reflected pulse suggests that a similar state of affairs exists at all distances from the vertex, but the actual calculations only extend to a plane section whose distance from the vertex is four times the focal length. The unsatisfactory convergence of the series precludes the investigation of subsequent changes in the distribution of pressure. It is finally pointed out that these results apply to a certain extent to a finite parabolic mirror.

1. The postulation of a multiple curve for primals of sufficiently large order in space of any number of dimensions has been obtained recently by J. A. Todd, by a simple and elegant degeneration argument which, however, is not deemed to be a conclusive proof by the author himself. And, indeed, in order to make sure of the unconditional validity of such an argument, one should ascertain whether

(i) the postulation θk of an irreducible non-singular curve ϲ, of order c and genus p, for the primals of sufficiently large order n of [r + 2] (r ≥ 1), required to go through it with multiplicity k (≥ 1), is a function of k, c, p, n, r only;

(ii) it is possible, by means of a continuous variation of ϲ, to reduce this curve to connected polygon ϲ′ having the same virtual characters as ϲ, in such a way that each intermediate position of ϲ is still irreducible and non-singular;

(iii) the postulation θk of ϲ equals the similarly defined postulation of ϲ′.

The differential equation of wave propagation is

where u is a variable describing the state of a medium, x, y, z are rectangular coordinates, and r is the time (the time scale is assumed to be such that waves are propagated with unit velocity).

13. The main object of this paper is to supply simple algebraical proofs of the general formulae for the number of particles, at any time t, of the rth product in a series of successive radioactive transformations. This is done both for ‘Case 1’, where the matter is initially all of one kind (paragraphs 3 and 4), including a stable end-product (paragraph 5), and for ‘Case 2’, where the successive products are initially all in existence and in equilibrium (paragraph 6). In paragraph 7 it is remarked that there exists a relation between the amounts of the end-product in ‘Case 1’, supposed to be the (r + 1)th product, and of the preceding rth product in ‘Case 2’, for the same time t, and the necessity of this relation is explained on general grounds. In paragraph 9 numerical values are given for the particular example (paragraph 8) considered by Rutherford in his Newer alchemy (pp. 11, 12), whilst paragraphs 10 and 11 deal with the degree of accuracy attained, and show the need for close accuracy in the experimental data.

There are two main anomalies in the spectrum of O++, for which so far no satisfactory explanation has been given: (1) the calculated values for the configuration (1s)2 (2s)2 (2p)2 of the ratio of the intermultiplet separations (1D−1S)/(3P−1D) is not in agreement with the observed value, (2) within the 3P term the ratio of the separations indicates a departure from Russell-Saunders coupling. We are here concerned with (1).

I. It is shown that a general argument can be given for omitting the diverging parts of any quantized field interacting with a particle. The field equations thus freed from all singularities still contain the reaction of the field on the particle to the same extent as the reaction can be derived classically from the conservation of energy. The new field equations can be solved generally in terms of a simultaneous set of integral equations and do not give rise to any fundamental difficulties whatsoever.

II. The new field equations are applied to the multiple processes of the meson theory which occur, for instance, when a meson collides with a nuclear particle, the meson splitting up into a number n of secondary mesons. The cross-sections γn are worked out for n = 1, 2, 3, 4. γn has a maximum at some energy εn which increases with n. For ε ≫ εn, γn decreases like ε−2n−4 except for n = 1, when γ1 ∞ ε−2. γ1 (ordinary scattering) is larger than all the other γ's at all energies, but the probability of high multiplicities is comparable with that for low multiplicities or even greater in some energy regions.

It is shown that, for crystals of the cubic system, the reciprocal of the integral breadth of a Debye-Scherrer line is cos θ/λ times the volume average of the thickness of the crystal measured at right angles to the reflecting plane. The result is applied to calculate the integral breadths of reflexions from crystals having the external forms of rectangular parallelepipeds, tetrahedra, octahedra and spheres. Except for spheres, the integral breadths are a function of the indices of reflexion as well as of the size of the crystal and the angle of reflexion. For cubes the variation with indices of reflexion is about 15%.

The division of ideal theory into the additive and multiplicative branches of the subject is a marked one of long standing. It is the object of this paper to carry the distinction a stage further, by showing how the group-theoretic methods of the additive branch can be extended to modules, that is, to Abelian groups, provided the ring of operators is commutative.

In a recent paper, Eddington raises an objection against the customary use of the Lorentz transformation in quantum mechanics, as for instance when applied to the theory of the hydrogen atom or the behaviour of a degenerate gas. This objection seems to us to be mainly based on a misunderstanding, and our purpose here is to show that the practice of theoretical physicists on this point is quite consistent. The issue is a little confused because Eddington's system of mechanics is in many important respects completely different from quantum mechanics, and although Eddington's objection is to an alleged illogical practice in quantum mechanics he occasionally makes use of concepts which have no place there. Such arguments will not have any bearing on the question whether or not the practice in quantum mechanics is logically consistent—although they may have bearing on which of the two systems describes physical phenomena better.