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.

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%.