1. This note originates from a question put to us by Mr W. R. Dean concerning series of the type

where θ is irrational. Series of the type

are familiar: it is well known, for example, that

may have any radius of convergence from 0 to 1 inclusive, according to the arithmetic nature of θ. It is natural to ask how this radius is connected with that of

and, more generally, how those of (1·1) and (1·2) are connected.

1. Let (a, b) denote an unbounded interval. Given the numbers s0, …, sn−1, consider the problem of finding what conditions they must satisfy in order that there shall be a function f(x), 0 ≤ f ≤ 1, such that

In my paper under the same title, to which I shall refer in future as I‡, I generalized the Vitali covering principle from the case of Lebesgue measure to the case of any non-negative additive function. This allowed me to establish the relative differentiation of additive functions. The convergent sequences of sets in this generalized form of the covering principle were restricted to sequences of concentric circles, and therefore the differentiation arrived at was that in the symmetrical sense. In the present paper, I extend the principle to the case of any regular convergent sequences of covering sets; and then establish the relative differentiation of additive functions in the general sense, and in particular the differentiation of indefinite integrals with respect to any measure function. This problem has a complete solution. It is established that indefinite integrals are differentiable at almost all points. In the case of the general measure function, it is not true that the derivative is equal to the integrand at almost all points, but necessary and sufficient conditions are given under which this is true.

1. The only double binary forms for which the complete systems of invariants and covariants are known appear to be the (1, 1), (2, 1) and (2, 2) forms, for which the complete systems were determined sixty years ago by Peano. In the present paper we determine the complete system of the binary (3, 1) form and establish its irreducibility. The system proves to contain twenty forms, and is thus scarcely more complicated than that of the (2, 2) form which includes eighteen concomitants, and much simpler than that of the (2, 1, 1) form derived by Gilham. It is hoped in a subsequent note to interpret the system geometrically.

1. A set of functions {øν (t)} (ν = 1, 2, …) is said to be closed L2 in (a, b), if

implies f (t) = 0 p.p. in (a, b). It is well known that a set of functions is closed L2, if and only if every function of L2 (a, b) can be approximated in the mean square sense as closely as desired by finite linear combinations of the øν (t).

A curve Γ has been defined as a normal multiple of a curve C if it contains an involution In, of order n, which has the properties: (1) the sets of In are in (1–1) correspondence with the points of C; (2) each set In consists of n points which are distinct in the birational sense; (3) In is generated by an Abelian group G of order n of birational transformations of Γ into itself. We shall denote the field of complex numbers by k, the function field of C by k(C), and the function field of Γ by k(Γ). Conditions (1) and (3) imply that k(Γ) is a commutative normal (i.e. Galois) extension of k(C). The object of this note is to show how, given the curve C and the Abelian group G of order n, we can construct a curve Γ with the properties (1), (2), (3).

In this paper formulae are developed for the first and second focal lengths, and the positions of the first and second principal planes of a type of electrostatic lens which has been the subject of study (mostly experimental) in several previous papers. The lens, which is commonly used in electron optical devices, lends itself to a theoretical study, although this does not appear to have been attempted before. It consists of two equal semi-infinite cylinders placed end to end so that their axes coincide and the ends are separated by a small gap. If the cylinders are at potentials V1 and V2 and we write σ = V2/V1, the system behaves as an electron lens when σ > 0 and as an electron mirror when σ < 0. In the latter case some experimental results have been given by Nicoll(1) who also studied the focusing action in the case σ > 0 and, in particular, the formation of intermediate images when σ ≪ 1 and when σ ≫ 1. But for the precise formulation of the relationship between σ and the number of cross-overs a theoretical study, based on the paraxial equation, would be necessary. The problem will be indicated below. An experimental determination of the lens characteristics for values of σ from about 2 to 15 and for several gap widths has been made by Spangenberg(2), whose results will be compared with those obtained in the present paper. The two-cylinder lens has also been studied by Klemperer and Wright(3) using an experimental and a numerical (trigonometrical) method, and some crude analytical results have been given by Gray(4).

1. In this paper we shall show that the complete system of concomitants of a binary (4, 1) form in two sets of binary variables (x0, x1), (y0, y1) consists of 63 forms. The methods employed are those of a previous paper.

Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define

1. The Petzval field curvature produced in a compound glass optical system of axial symmetry is given by a well-known formula. If the system consists of a number of media, of refractive indices n0, n1, n2, …, having spherical faces whose radii of curvature are r0, r1, r2, …, the formula is [see for example, (1) or (2)]

and this is in common use amongst those concerned with the design of optical equipment. The analogous integral expression in the electron optical case has not, however, received the attention it deserves, in spite of the developments in electron optics during the past decade. In 1935 Glaser(3) presented the third-order error theory of an axially symmetric electron optical system.

1. The reflexion of a simple harmonic train of surface waves by a rigid plane barrier inclined at an angle α to the undisturbed free surface of the water is considered in the particular case in which 2sα = π, where s is an integer. The solution if s = 2 has been given by Gourret(1), while the solution if s = 1 is well known; the solution for any integral value of s is given here.

1. A paper with the same title was published with Professor H. F. Baker in 1920; that paper was severely compact and would seem to deserve elucidation, more detail, and some revision. We use a symbolism which, while leaving the proofs geometrical, readily converts itself into ordinary analysis, as we shall show in § 12.

1. In a recent paper(1) expressions were found for the elastic stresses produced in a semi-infinite elastic medium when its boundary is deformed by the pressure against it of a perfectly rigid body. In deriving the solution of this problem—the ‘Boussinesq’ problem—it was assumed that the normal displacement of a point within the area of contact between the elastic medium and the rigid body is prescribed and that the distribution of pressure over that area is determined subsequently. The solutions for the special cases in which the free surface was indented by a cone, a sphere and a flat-ended cylindrical punch were derived, but no attempt was made to give a full account of the distribution of stress in the interior of the medium in any of these cases.

If (a1, b1, c1), (a2, b2, c2), (a3, b3, c3) be three sets of numbers subject to the six relations such as and with a positive determinant, then the four tetrahedra, M, M1, M2, M3, whose angular points have coordinates given by the rows of the four matrices

are such that every two are mutually inscribed. The sixteen points so arising are the nodes of a Kummer surface having for one form of its equation

The Eulerian equations obtained by varying a Lagrangian containing certain field quantities qα and their derivatives of any order are cast into canonical forms. A modified canonical form for the equation of motion is given when the conjugate variables are not independent. Finally a special case is discussed.

The success of Stoner's extensions of the theory of collective electron ferromagnetism(1) opens up new possibilities of considerable importance. The background of the theory can be found in standard works on the theory of metals(2) and statistical mechanics(3) and will not be given here.

The paper considers the application of Dirac's classical theory of radiating electrons to examine the motion of an electron in a uniform magnetic field. The non-relativistic equations of motion are solved exactly. It is shown that for the particular case when the motion is confined to a plane, the physical motion is such that the electron describes an equiangular spiral with velocity which steadily decreases as

exp {negative constant × t};

and in the non-physical motion the electron spirals outwards with velocity which steadily increases as

exp {positive constant × t},

and ultimately the electron escapes to infinity. The relativistic equations of motion are solved approximately as a series in ascending powers of the field. It is shown that in the physical motion the velocity begins to decrease, and hence after a time the motion is given correctly by the non-relativistic solution.

I take this opportunity to express my deepest gratitude to Prof. Dirac, who suggested this problem, for his patient guidance and supervision.

In this paper we consider the properties of a point-curve correspondence between two (generally distinct) surfaces F and G, paying special attention to the idea of ‘induced’ and ‘extended’ correspondences. We also outline the theory of point-curve corre spondences between a curve and a surface, which is not only of interest in itself, but is of importance because such correspondences arise, as do correspondences between curves, as induced correspondences of point-curve correspondences between two surfaces. The results depend chiefly on Lefschetz's theory of correspondences between algebraic curves(1) and the properties of intersections of cycles on an algebraic surface given by the author (2), as well as on the preceding paper of this series (3). Use is also made of a paper by Severi (4) on the self-correspondences on a variable curve of an algebraic surface. The theory of correspondences on a single surface will be further developed in the next paper on this subject.

1. Introduction. Recent experiments by Amaldi and his collaborators on the scattering of high-energy neutrons (of 10–15 MeV.) by protons(2) have disclosed a considerable anisotropy in the angular distribution of the scattered particles. Theoretical discussions of this problem show an interesting feature in that the results depend sensitively on the basic assumptions involved with regard to the charge dependence of the neutron-proton interaction. This can be seen in particular from calculations by Rarita and Schwinger(3) and by Ferretti(4). The former authors started from the assumption of a distance dependence of this interaction represented by a square well, while the angular and spin dependence included terms of the axial dipole type. If the charge dependence was further assumed to be of the ‘symmetrical’ type, they found a value for the anisotropy in strong disagreement with experiment, whereas the total cross-section agreed with the measured value; a ‘neutral’ theory, on the other hand, yielded agreement as regards anisotropy, but a total cross-section too large by a factor of the order of 1·5. Ferretti investigated the scattering on Bethe's neutral meson theory(5) and found satisfactory agreement with regard to both angular distribution and total cross-section. It should be stressed that all calculations mentioned were performed in the approximation in which only the contributions of the S- and P-waves are considered.

In Part I of this series it was shown that in a metal with overlapping energy bands, one of which is nearly full and the other nearly empty, there exists a critical temperature below which spontaneous magnetization will be present, Tc being given approximately by the equations

where kθ is the exchange energy term postulated by Stoner, b is the ratio of excess vacancies in the d-band to the maximum possible number of parallel spins, and ζd0 is the equivalent energy depth of the vacant levels in the d-band.