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.

The expression

for the number of linearly independent line complexes of degree n in space of r dimensions was derived by Sisam as a particular case in the determination of the number of linearly independent hyperconnexes of given degrees. The purpose of this note is to generalize Sisam's result by determining the number of linearly independent algebraic forms of a special type to which we give the name k-connex.

Throughout this paper we shall suppose that denotes a set of elements x in which a Lebesgue measure is defined and that itself is measurable and has finite measure. A (1, 1) transformation T of into itself is called an equimeasure transformation if the transform T E of any measurable subset E of is measurable and has measure equal to that of E. Then, if f(x) is integrable in , it is plain that f(Tx) is also integrable and that

1. The object of this paper is to discuss conditions of validity of the Parseval formulae for Fourier integrals:

where the transforms are defined by ordinary convergence; we shall not be concerned with the more elegant theory in which they are given by convergence in mean.

A general theory of ‘strength’ for Hausdorff methods T of summability was given in the first part of this paper, to which we shall refer in the following as I. Those methods T which are regular (that is, at least as strong as ordinary convergence C), are defined by the linear transforms

of a given sequence (sk). Here ø(t) (that is, each of its real and imaginary parts) is of bounded variation (b.v.) in the closed interval 〈0, 1〉 [see I, (4)]. The additional condition for consistency (with C) is that ø(t) is continuous at t = 0, and that ø(1) − ø(0) = 1 [I, H. 3]. We write also T ˜ ø(t).

Let f(t) be defined and measurable for t > 0, and suppose that it has non-vanishing moment constants

where the integrals are Cauchy-Lebesgue integrals, or possibly if f(t) is null for t > t0 but not for t < t0 (a convention to which we adhere throughout).

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.

1. Definition 1. A linear set E is said to possess the property C if, to any sequence of positive numbers {ln} (n = 1, 2, …), there corresponds a set of intervals, of length not greater than l1, l2, …, which includes all the points of E.

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).

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

We shall consider plane Borel sets of points, though the result and the proof hold in space of any number of dimensions and for more general sets.

The aim of this paper is to give a general theory of the ‘strength’ of Hausdorff methods of summability. These methods are defined by the linear transforms

of a sequence (sk). Here the μk form a given sequence of real or complex numbers and the Δp(μk) denote their differences of order p; i.e. Δ0(μk) = μk and

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.

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′.

1. There are several known types of expansions of Mathieu functions, i.e. mod 2π periodic solutions of Mathieu's equation ((9), chap. 19),

The simplest expansion is the Fourier series

Almost equally well known are Heine's expansion ((4), p. 414; see also (5) and (6))

and

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%.

1. I appreciate the careful reply which Dirac, Peierls and Pryce (hereafter referred to as DPP) have made to my criticism of the usual theory of the eigen-functions of a hydrogen atom. Their paper will be generally welcomed because there has not, I think, previously been available an authoritative and clear statement of what is assumed and the grounds for assuming it. Their defence, which is partly on lines I had not expected, has required very serious consideration. But the result of making the arguments more explicit is two-fold. The theory is perhaps more self-consistent than it had appeared to be; but on the other hand, the pressing need for amendment becomes too plain to be overlooked. I have endeavoured to show in the last twelve years that this amendment opens out very fertile extensions.

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).

In the course of the development of a special method of interpreting thermo-dynamic data in terms of a generalized force field for the mutual interactions of the molecules concerned, expansions were required for integrals of the following general type, which apparently have not hitherto been discussed. We require explicit expansions of F(α, s, n), in terms of the parameter α, for the integral values 0, 1, 2, 3, … of s and n, where