The object of this paper is to draw attention to a series of five types of rational quartic primal in [4]. Two of these are already known. The greater part of this paper deals with one of the other three types of primal and with a new symmetrical quarto-quartic Cremona transformation of [4] determined by a homaloidal system of primals of this type passing through a certain surface of order nine. It is hoped in a subsequent paper to continue the investigation into the remaining two types of primal.

On a surface F′ an algebraic self-correspondence T of period n defines a cyclic involution In of sets of n points. Then if there exists a surface F whose points are in (1, 1) correspondence with the sets of In, the surfaces F, F′ will be said to be in (1, n) cyclic correspondence. The purpose of the present paper is to show that, when n is a prime number and with certain restrictions upon the united curve of the self-correspondence T, the irregularities of the surfaces F and F′ are equal.

The Gliding of a Plate on a Stream of Finite Depth*

Page 597. The line after equation (44) should read

Proc. Cambridge Phil. Soc. 31 (1935), 589–603.

The modified field equations proposed by the author can be derived from the energy density

(where is the electric displacement, the magnetic induction, the Poynting vector) with the help of the commutation laws for these vectors.

The object of this paper is to make some remarks about a minimal problem suggested by the theory of the buckling of plates. A question left open in my previous paper on this subject is thereby solved.

We shall discuss the annihilation of positrons by a process in which only one quantum of light is emitted. The positive electron is regarded as a hole or unoccupied state in the distribution of electrons in negative energy states, and is destroyed by an atomic electron jumping down into it with the emission of a single quantum. This process is only possible in the presence of a nucleus, which can take up the momentum liberated, and is therefore most probable when the atomic electron is a K-electron.

7. We have discussed the gliding of a plate of finite length on a stream of finite depth. Numerical calculations have been made for the case when the angle of incidence of the plate to the stream was 30°, the results for any other angle being similar.

It was found that, for a given depth of stream and a given height of the trailing edge above the bed of the stream, the value of the lift increases with the length of the plate, until finally, when the plate is infinitely long, the lift assumes a maximum value. Further, for a given depth of stream, the total normal lift on the plate is independent of its height above the bed of the stream, when the length of the plate is small, except when the trailing edge of the plate is above the surface of the stream. Finally, when the depth of the stream is very large and the plate is near the middle of the stream, then our solution approximates to the classical Rayleigh flow past a plate in an infinite fluid.

The method of determining the focal length f of a thin converging lens by finding the minimum distance Δ0 between image and object is well known. When the thickness of the lens is negligible, f = ¼Δ0.

In the general coaxial system, the media at the two ends have refractive indices μ1, μ2, and there is a finite distance t between the principal or unite plances. The two focal lengths are unequal, since, f1/f2=μ1/μ2, and the nodal points do not coincide with the principal points.

It is well known that the boundary value problem for the non-linear differential equation

can be reduced with help of a Green's function K (x, ξ) to a non-linear integral equation of the type

There is a theorem for three circles in a plane, that if three tangents, each of two of these circles, either all transverse or one transverse and two direct, can be drawn to meet in a point, then the three tangents, each of two of the circles, respectively conjugate to those first taken, likewise meet in a point. The theorem was stated by Quidde, with a proof for the necessity of the condition as to the tangents to be taken, in a paper designed to establish Steiner's solution of Malfatti's problem. Casey gives the theorem with omission of the condition for the character of the tangents, as does Salmon, who, however, gives a proof depending on the right choice of certain square roots which enter. Quidde's theorem is stated, accurately, in the Nouvelles Annales, and a simple metrical proof, from the diagram drawn (essentially Quidde's, see 6 below) is given later in the same Journal by Mannheim; this is practically repeated by Hart. Recently, Prof. Neville, emphasizing the necessity of the condition for the character of the tangents, has called attention to Quidde's paper.

1. Given a sequence of variables a1, a2, …, we define a sequence of polynomials B1, B2, … by

For every positive integer n, Bn is a polynomial in a1, …, an, with rational coefficients. We write Bn ≡ Bn (a1, …, an) to indicate the variables on which Bn depends. From the polynomials Bn, we form a new sequence of polynomials Dn in accordance with the following definitions.

Let us consider a discontinuous bivariate distribution. That is, let us consider N × Q non-negative values pνq (ν = 1, 2, …, N; q = 1, 2, …, Q), being the theoretical probabilities of the νth value of a variate Xμ (μ = 1, 2, …, N) concurring with the qth value of a second variate Ys (s = 1, 2, …, Q).

The following paper is a study of abstract algebras qua abstract algebras. As no vocabulary suitable for this purpose is current, I have been forced to use a number of new terms, and extend the meaning of some accepted ones.

Experiments are described comparing the effects of neutrons slowed down in paraffin wax at the temperatures of liquid nitrogen and liquid hydrogen with those obtained at ordinary temperatures. It is found that the absorption produced by certain substances increases as the temperature is lowered. The transformations produced in these substances also increase, but generally to a smaller extent, probably owing to more neutrons being absorbed in the paraffin: the exact figure depends on the thickness of the layer of cooled paraffin.

A few experiments were also made substituting liquid hydrogen for paraffin wax (without change of temperature). The effect of this also appears to depend on the thickness of the layer, but it is not yet possible to draw definite conclusions.

The probability relations which can occur between two separated physical systems are discussed, on the assumption that their state is known by a representative in common. The two families of observables, relating to the first and to the second system respectively, are linked by at least one match between two definite members, one of either family. The word match is short for stating that the values of the two observables in question determine each other uniquely and therefore (since the actual labelling is irrelevant) can be taken to be equal. In general there is but one match, but there can be more. If, in addition to the first match, there is a second one between canonical conjugates of the first mates, then there are infinitely many matches, every function of the first canonical pair matching with the same function of the second canonical pair. Thus there is a complete one-to-one correspondence between those two branches (of the two families of observables) which relate to the two degrees of freedom in question. If there are no others, the one-to-one correspondence persists as time advances, but the observables of the first system (say) change their mates in the way that the latter, i.e. the observables of the second system, undergo a certain continuous contact-transformation.

1. The purpose of this note is to prove the uniqueness of the solutions of two closely related differential equations, under suitable boundary conditions. The first is the equation satisfied by the potential in the electrostatics of the new field theory proposed by Born, namely,

where øx stands for etc.

The theoretical method for computing the grade of a curve on an algebraic surface is well known. In practice difficulties arise which are not considered in the theory; so that it seems worth while to describe a practical method. This is done in § 1 of this paper. The method is then applied to some examples with the object of discovering whether Noether exceptional curves are necessarily exceptional curves†. In particular, a certain quintic surface with three tacnodes is studied, and our examination leads us to results which differ from those which have been accepted up till now. Another example illustrates the limitations of a practical method for computing grades, because of the possible presence of infinitesimal curves, and leads to the transformation of the quintic surface with two tacnodes into a double plane of order ten of a certain type, which has the singularity known as a (5, 5) point on its branch curve. Light is thrown on the Noether composition of this singularity by the transformation, which also shows the relation between two well-known types of surface for which the Noether relation p(2) = p(1) − 1 does not hold.

In a previous paper the present writer gave a solution of the problem of determining the circulation around a thin elliptic cylinder in a steady stream of slightly viscous fluid when the major axis of the cylinder is inclined at a small angle to the direction of the flow at infinity. The present note gives a largely qualitative analysis of the manner in which this circulation is built up when the motion of the fluid or cylinder is started instantaneously from rest with a given velocity.