This paper is a sequel to one recently published, in which the regularity of surfaces was investigated by means of certain criteria suggested by Comessatti. We suppose throughout that the surface F under consideration is of general type in Sr, and that it has curve sections C of order n and genus π. It may be recalled that in the previous work the two most useful criteria were the following theorems:

If π < ½ (n + 2), then F is regular or referable. (I)

If πr (n) ≤ π ≤ n, where πr (n) is the maximum genus of a Cn in Sr, then F is regular or referable. (II)

Castelnuovo has shown that the maximum freedom of a linear system of curves of genus p is 3p + 5, and that a system with this maximum freedom consists of hyperelliptic curves which can be transformed into a system of curves of order n with an (n − 2)-ple base point and a certain number of double base points; the only exceptions being that when p = 3 the system may be transformable into that of all quartics, and when p = 1 the system may be transformable into that of all cubics. Further, since, in the transformed system, the characteristic series is a it is non-special and hence the redundancy of the base points is zero, therefore each of the double points of this system reduces the freedom by exactly three; and hence if we remove all the double points we get a system of curves of genus p′ and freedom r′ = 3p′ + 5 with only one base point. If we take this point for origin the system of curves can be represented by a single Newton polygon containing in its interior exactly p points, collinear since the curves are hyperelliptic (p ≠ 3), and containing on its boundary 2p + 6 points. From this we can deduce immediately a theorem concerning convex polygons drawn on squared paper; I shall now give an a priori proof of this theorem.

1. If the canonical series of an algebraic curve of genus p is compounded of an involution of sets of points on the curve then the involution must be rational and of order two, and the canonical model is a repeated rational normal curve of order p − 1 with 2p + 2 branch points. An analogous question suggests itself for algebraic surfaces. Under what conditions is the canonical model of a surface, whose geometric genus is not less than two, a multiple surface, and what in this case are the properties of the simple surface on which the multiple surface is based? In this paper we give particular examples of multiple canonical surfaces and attempt to go some way towards the solution of the general problem.

It is familiar that a “double plane”, i.e. an algebraic surface conceived as consisting of each point of a plane counted twice, is unambiguously determined when its “branch curve”, or locus of points in the plane which count as two coincident points of the surface (the general point of the plane counting as two distinct points), is assigned; the only condition which must be satisfied by this curve being that it is of even order. In fact, if f(x, y) = 0 is the Cartesian equation of the curve, the double plane can be regarded as the limit of the surface

where A is a large constant, or as the projection of this surface, for finite A, from the point at infinity on the z-axis. If, however, f = 0 is of odd order, it is easily seen that the line at infinity also forms part of the branch curve of the double plane. A repeated portion of the branch curve is ineffective, for if φ (x, y) = 0 is any other curve, the surface

is birationally identical with the former merely by putting

The general conception underlying the following analysis is that of a field of potential which is invariant under a group of geometrical transformations. The boundary is to consist of a number of parts which transform into each other and the boundary values also transform into each other. To satisfy the conditions we build up functions which are invariant under the same group of transformations and combine them to give the prescribed boundary values over one section of the boundary. The boundary conditions on the other sections are then automatically satisfied. When the groups are simply translations or rotations we get functions periodic in either a Cartesian or an angular coordinate.

The geometrical or vectorial representation of a sample as a vector with n mutually perpendicular components corresponding to the n observations in the sample was introduced into statistics by Fisher (1), and led to the solution of many theoretical problems of statistical distributions. Subsequently Fisher (2) gave an alternative algebraic method applicable to finding distributions in connection with regression and the analysis of variance. The actual use of symbolic vector notation in deducing some of the properties of the sample vector—besides tending to stress the complementary character of these two different methods of proof, and the common principle underlying the analysis of any sample into its components—indicates also some points in connection with the assumption of the normal law in which it seems almost essential to retain the geometrical side of this vector representation. Moreover, it will be seen that the vector theory used here in reviewing briefly the analysis of a sample of one dependent variate can readily be extended to cover the case of correlated variates.

This article deals with the currents set up in thin-walled cylinders by two-dimensional field variation in the plane of the section, and is applied particularly to the problem of the distribution of alternating currents in strip conductors, a matter of some technical importance.

Boron and carbon activated by diplon and proton bombardment have been used as positron emitting sources in experiments on the annihilation radiation.

By coincidence counting with two Geiger-Müller tubes it has been established that the radiation is emitted in pairs.

By the coincidence method of Becker and Bothe it has been shown that the annihilation radiation consists only of “soft” quanta.

By ordinary absorption measurements it has been shown that the annihilation radiation is homogeneous with a hardness corresponding to 0·5 million e-volt.

The hydrogen discharge has for some time been established as a convenient source of continuous spectrum in the ultra-violet. The continuum extends from nearly 4000 Å. to the limit of a quartz spectrograph and provides a very good background for absorption measurements. A number of tubes have been described for producing this spectrum, but the following advantages are claimed for the form here described.

The object of the present paper is to report upon a case of neutron-produced disintegration of a type which has not previously been observed. The nucleus disintegrated was that of carbon and the peculiarity of the disintegration lies in the fact that three heavy particles resulted from the transformation. Hitherto no example of such disintegration resulting in more than two heavy particles has been obtained, and with carbon, in particular, several experiments agree in showing that this more usual type of disintegration is very rare indeed. In similar circumstances, and with neutrons of less than 12 × 106 electron volts energy, disintegration phenomena occur in oxygen or in nitrogen at least ten times as frequently as in carbon. Thus Harkins, Gans and Newson obtained only two examples of disintegration in 3200 pairs of photographs taken with a source of radiothorium and beryllium and an expansion chamber filled with ethylene, and concluded that in each case an atom of oxygen or some other impurity must have been involved. Likewise, Feather, in 2210 pairs of photographs with the neutrons of polonium-beryllium and an expansion chamber filled with a mixture of acetylene and helium, found only one example of a disintegration which could reasonably be ascribed to the nuclear reaction,

1. The steady motion of an incompressible viscous fluid, due to an infinite rotating plane lamina, has been considered by Kármán. If r, θ, z are cylindrical polar coordinates, the plane lamina is taken to be z = 0; it is rotating with constant angular velocity ω about the axis r = 0. We consider the motion of the fluid on the side of the plane for which z is positive; the fluid is infinite in extent and z = 0 is the only boundary. If u, v, w are the components of the velocity of the fluid in the directions of r, θ and z increasing, respectively, and p is the pressure, then Kármán shows that the equations of motion and continuity are satisfied by taking

A method is described for measuring the heat of adsorption of hydrogen on a clean tungsten filament. The value obtained is 2·8 × 104 calories per mol.