Nemilov and Pisarevskii (1957) have reported a 390 keV γ-ray in coincidence with neutrons from a Po-Li source, and interpret this as evidence for a level in B10 at 390 keV. A search for confirmatory evidence of this γ–ray has been unsuccessful, and it is estimated that such n–γ coincidences would have been detected in the experiment here described if they involved 8 per cent or more of the neutrons from the Po-Li source.

This paper is concerned with the theory of the probability distribution of the total number of electrons in the avalanche produced by the release of a single electron in the gas of a proportional counter. The disagreement between existing theory and experimental results is discussed and a new theory is proposed, based on the fact that fluctuations in the number of electrons at a given point in the avalanche are accompanied by fluctuations in the average electron energy. This aspect of the problem is incorporated directly into a simple one-dimensional model of the multiplication process, and the resulting distribution function has a mathematical foim in agreement with that observed experimentally.

The fluctuation in the number of electrons predicted by this theory is not constant, but is determined by a parameter which, for large mean values, is essentially the mean fraction of electrons in the avalanche having energies above the ionization energy of the counter gas. Limits on the variation of this parameter are obtained by calculations of the mean values using a particular two-dimensional model, in which electrons are divided into two classes according as they have energies above, or below, the ionization energy. The experimentally observed fluctuation lies within the predicted range and close to the lower limit; it is concluded that there is little scope for improvement in the resolution to be obtained from the conventional type of proportional counter.

Let R be a ring and let ΓR be the Jacobson-Perlis radical. It is shown that the radical of the ring of row-finite matrices over R is the ring of row-finite matrices over ΓR if and only if ΓR is right-vanishing. This is done by extending the results of an earlier paper. One interesting consequence of the theorem is noted.

In previous papers [1, 2] the author has considered the stability of a current-vortex sheet in a non-diffusing incompressible fluid, the magnetic field being parallel to the plane of discontinuity. In [1] a criterion was given for a parallel magnetic field to stabilise a vortex sheet, and in [2] the energy balance of this system was considered and it was shown how the magnetic energy level is increased at the expense of mechanical kinetic energy when the system becomes unstable. In this paper the oscillations on a plane interface are considered when the magnetic field is not parallel to the interface.

Let S be a point set of the Euclidean plane, such that

(i) S is bounded,

(ii) the closure of S has unit Lebesgue measure.

Let P be an arbitrary set of n points contained in S, and let l(P) denote the total length of the shortest system of lines connecting the points of P together. Define ln to be the supremum of l(P), taken over all sets P of n points in S. Beardwood, Halton, and Hammersley [1[ proved that there exists an absolute constant α, independent of S, such that

An expression is found for the biharmonic Green's function, G(x, y; x1, y1), for an infinite area in the x, y-plane bounded internally by a single curve; at all points of the boundary G has a zero of the second order.

A method is developed for finding the distribution of velocity, density, pressure and magnetic field behind an expanding strong cylindrical shock wave in an infinitely conducting fluid in the presence of a magnetic field.

In the flow of such a fluid there are two distinct methods for producing a strong shock:

(i) by imposing the usual density ratio across the shock, as in the non-magnetic cnse, and

(ii) by imposing a large magnetic field, such that the Alfvén velocity is very much larger than the speed of sound. The distribution of the various physical quantities and the velocity of propagation of the shock are discussed for both cases, and numerical results given.

Let p be a prime and let F be a polynomial in one variable with coefficients in GF(p), the field of p elements. Let d be the degree of F, and let r+1 denote the number of distinct values F(µ) as µ. ranges over GF(p). A generalization of the Waring problem modulo p leads to the question the determination of a lower bound for r.

In an earlier paper (cf. [1]) I had given a generalization of the concept of an absolute discriminant to arbitrary finite number fields K as base fields. In a second paper (cf. [2] 2. 3, see also [3] 1. 3) it was shown that the discriminant δ(Λ/K) of a finite extension Λ of K determines the structure of the ring ς of algebraic integers in Λ qua module over the ring ο of algebraic integers in K. The purpose of the present note is to establish a corresponding result for an arbitrary Dedekind domain ο, and finite separable extensions Λ of its quotient field K. The general theory of discriminants and module invariants developed in [1] and [2] for algebraic integers applies in principle to arbitrary Dedekind domains, as already pointed out in the earlier papers. It is usually evident what further hypotheses—if any—have to be imposed to ensure the validity of any particular theorem. For the quoted result of [2] this is, however, not at all clear. In fact the proof involves the proposition:

I. If an element of K is a square everywhere locally then it is a square in K.

In a recent paper [1], we have given some account of theories of equivalence and intersection on a singular algebraic surface and have shown that such theories share many of the simple properties enjoyed by corresponding theories on non-singular surfaces. Another paper [2], now in preparation, will extend this work to singular varieties of arbitrary dimension. In the meantime, Zobel [3] has drawn attention to some suspect arguments of Samuel [4] concerning the specialization of intersections on a singular variety.

In ring theory there is the following theorem (cf. [1], p. 39):

If R is a ring satisfying the descending chain condition for left ideals, then the following three conditions are equivalent: (i) R is primitive, (ii) R is simple, and (iii) R is isomorphic to the complete ring of linear transformations in a finite dimensional vector space over a division ring.

The purpose of this paper is to call attention to a very simple example of intersections on a singular variety, and to its effect on intersection theories relating to ambient varieties with singularities; in particular it will be shown how the example—to which we refer throughout as Example A—invalidates an aspect of the theory put forward by P. Samuel ([1], Ch. VI).

Let m, n, q denote positive integers, p a prime, and a, b, h, r, s, t, u, v integers. If (r, q) = 1, let [r, q] be the integer s for which 0 < s ≤ q and rs ≡ 1 (modq). Let

It is shown that the greatest value of the resultant shear in the Saint-Venant torsion problem for an aeolotropic material possessing digonal elastic symmetry occurs on the boundary of the cross-section.

In a recent paper (Shail [1]) the present author considered the problem of finding a two-centred expansion of the retarded Helmholtz Green's function This work formed an extension to that of Carlson and Rushbrooke [2] (also Buehler and Hirschfelder [3]) on the Coulomb Green's function and arose out of considerations of the interaction energy of two charge distributions taking account of electromagnetic retardation. The two-centred expansion obtained in [1] took the form of a double Taylor series, each term being interpreted as a Maxwell multipole—multipole interaction energy between two “charge” distributions coupled through a retarded scalar field. The first few terms in the expansion were also given in spherical polar coordinates.

A solution is obtained by real variable methods for a Cauchy problem for the generalised radially symmetric wave equation. A solution of this problem has been given by Mackie [1] employing the contour integral methods developed by Copson [2] and Mackie [3] for a class of problems occurring in gas dynamics. The present approach employs a simple definite integral representation for the solution and reduces the problem to solving an Abel integral equation. The real variable approach avoids the unnecessary restriction of the initial data to be analytic and also avoids the difficulty encountered in the complex variable approach in continuing the solution across a characteristic. The solution is in fact obtained in a form valid everywhere in the region of interest.