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.

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.

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

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.

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.

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.

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.