This paper is concerned with the extension to [2n] of a well-known symmetrical configuration in [4], namely, that of three rational quartic curves, with six points in common, having the property that a trisecant plane of any one which meets a second also meets the third.

1. In the present note we obtain the postulation of a multiple surface of S4 for primals of sufficiently high order; the method is that previously used by Campedelli to determine the postulation of a multiple curve of S3.

The object of this paper is to make more precise the results given by Enriques-Campedelli for canonical surfaces, that is, surfaces whose prime sections are the canonical curves. We consider the case of regular surfaces in space of three dimensions (pg = pn = 4), and of regular multiple planes (pg = pn = 3).

A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.

The complete (or “extended”) symmetry groups, investigated in Part I, are certain groups of orthogonal transformations, generated by reflections. Every such group has a subgroup of index two, consisting of those transformations which are of positive determinant (i.e., “movements” or “displacements”). The positive subgroup (in this sense) of [k1, k2, …, kn−1] is denoted by [k1, k2, …, kn−1]′, and is “the rotation group” (or, briefly, “the group”) of either of the regular polytopes {k1, k2, …, kn−1}, {kn−1, kn−2, …, k1}; e.g., [3, 4]′ is the octahedral group.

1. Introduction and summary. The problem of the elastic stability of a simply supported rectangular plate, compressed by two equal and opposite forces acting in the plane of the plate (see Fig. 1), was first attempted by A. Sommerfeld, and later by S. Timoshenko. The former produced a solution which in a later paper he admitted to be liable to very considerable error, while the latter constructed a solution by means of the well-known strain-energy method. In many problems this method gives results in very close agreement with those obtained in a more rigorous manner, but, in the particular case considered here, it appeared likely that the error would be appreciable owing to the underlying assumption that the only stresses in the plate occurred along the common line of action of the two external forces.

In an excursus developing a modern view of the Carnot-Kelvin aspect of thermodynamics, appended recently to the letters from Clerk Maxwell to his friend W. Thomson reporting the early tentative evolution of the theory of the electro-dynamic field, the account presented by the writer stopped short at putting emphasis on the mystery of temperature, as a unique and supremely significant property of matter in bulk, with its trend to uniformity as presenting the basic thermal problem. The side of the subject involving dynamical analogy was there absent from the development, which was purely formal. It may be well, however, now to offer some general notions such as may be put on record, which arise naturally from that mode of approach to the subject.

The previous paper dealt first with the shot effect without secondary emission. We considered a stream of electrons arriving at random, with a fixed probable density N, at the anode of the resistance-capacity coupled first valve of a linear amplifier. This does not allow for the effect whereby the arrival of electrons alters the anode potential, and consequently the probability of arrival of succeeding electrons. The valve must be working with what is conventionally called an infinite internal resistance, and without space-charge effects. Each electron arriving at time tr is supposed to produce at time t an output effect f(t − tr) from the amplifier. We considered the hypotheses, first, that electrons may, with specific probabilities, have lives of any length on the anode system, during which they add − ε/C to the anode potential, and alternatively that each electron adds to the anode potential of the valve whose anode to earth capacity is C and whose feed resistance is R. These hypotheses determine the form of f(t − tr) when the behaviour of the main part of the amplifier is known. Since the amplifier is linear, the effects of various electrons are additive, so that the total output is ∂(t) = ∑rf(t − tr).

The type of integral considered, is of frequent occurrence in problems in the kinetic theory of gases and in particular is required in the paper which follows, in calculating viscosity. The result is an extension of one given by Berger, and the analysis follows closely the discussion of Gauss's formula for approximate quadrature in Whittaker and Robinson, The calculus of observations; the novelty lies in the use of Sonine polynomials, which are peculiarly well suited for the discussion of this type of integral.

A method of determining the coefficient of viscosity of a gas of spherically symmetrical molecules under ordinary conditions has been given by Chapman. His result is equivalent to

where m is the mass of a molecule of the gas, T is the absolute temperature, k is Boltzmann's constant 1·372. 10−16 and ε is a small quantity which later investigations on a gas in which the intermolecular force is inversely proportional to the nth power of the distance have shown to vary from zero when n = 5 to 0·016 when n = ∞ (equivalent to molecules which are elastic spheres); it may reasonably be supposed that ε is positive and less than 0·016 in all cases which are likely to be of interest, and it will be neglected in this paper. Also

π(r) being the mutual potential energy of two molecules (that is, the repulsive force between them is − ∂π/∂r), and r0 the positive zero of the expression in the denominator, or the largest such positive zero if there are several.

The theory of transport phenomena in metals depends upon the solution of an integral equation for the velocity distribution function f of the conduction electrons. This integral equation is formed by equating the rate of change in f due to external fields and temperature gradients to the rate of change in f due to the mechanism which produces the resistance. If this latter rate of change is denoted by [∂f/∂t]coll it happens with some mechanisms that

where f0 is the equilibrium distribution, and ℸ is the time of relaxation which does not depend on the external fields. When equation (1) is true, the problem is comparatively simple, but in general [∂/∂t]coll is an integral operator and it is not possible to define a time of relaxation and a free path. It is known that at high temperatures, such that (Θ/T)2 can be neglected, where Θ is the Debye temperature, a free path exists; but, in general, special methods have to be used to solve the integral equation.

1. The reflexion of X-rays from an ideal crystal would be independent of temperature if the particles forming the crystal were at rest. Because of the motion of the particles a temperature effect does exist. It can easily be seen that only the component of the displacement at right angles to the reflecting plane is of importance, since it is the path difference between such planes which matters. The complete theory has been worked out by Debye and by Waller. The result is that, owing to the influence of the temperature, the intensity is multiplied by a factor e−2m, where

Here and μs is the mean square of the component of the displacement (in a direction at right angles to the reflecting planes) due to a normal mode π is the glancing angle of incidence of the X-rays on the reflecting planes, and λ is the wave-length.

In his work on mercury, Benedicks (1) observed that the small temperature peak in the neighbourhood of the middle of the long tube containing the conductor was displaced in one direction or the other along the tube according to the direction of flow of the electric current by which the conductor was heated. Normally this effect increased with the cube of the current intensity, and was explained as due to Thomson heat. If the temperature was made as uniform as possible by means of controlled heaters at the ends of the tube, the effect became of opposite sign and increased linearly with the current. This was called the “electro-thermal effect” and was explained as due to an actual change of thermal equilibrium by the electric current, or the production of a temperature gradient by a flow of electricity.

1. A familiar device in the study of statistical distributions is to form the moment-generating function

where the bar denotes averaging over all values of the statistical variate x. The moments μr of x are the coefficients of αr/r!, and the derived coefficients in the expansion of K ≡ log M are termed the semi-invariants kr. In particular,

and, for the normal (Gaussian) law,

we have the simple formula

In a study of the properties of hydrogen on tungsten by the method of contact potentials the following points have been established:

(1) The contact potential of a 92% covered surface of hydrogen on tungsten against bare tungsten is 1·04 V., and the Richardson constants for such a surface are, approximately,

A = 30, b = 5·60 V.

(2) A film of deuterium is 20 m V. positive relative to a similar hydrogen film.

(3) Over the range of temperatures and pressures used by Bryce in a study of the production of atomic hydrogen the films are nearly saturated, so that the production of atomic hydrogen is primarily due to a molecule striking a bare tungsten atom, one atom being adsorbed and the other going into the gas phase.

(4)The condensation coefficient for hydrogen molecules on cold tungsten is 0·01.

(5) The effective dipole moment of each hydrogen atom on the surface is −0·42 Debye unit and is independent of the fraction of the surface covered.

1. The cross-flow type of heat interchanger is shown diagrammatically in Fig. 1. One of the fluids involved is arranged to pass through a nest of small tubes, while the other streams past the tubes at right angles to them. To fix ideas we suppose that the tubes contain cooling water which enters at a temperature T1. The temperature of the water leaving a tube depends upon its position, but the water from all the tubes is mixed and is led to a single exit pipe in which the temperature is T2. In the same way the fluid to be cooled enters at a temperature t1 and leaves at a temperature t2. If the cooling water is not used again by being led through another nest of tubes also exposed to the fluid to be cooled, the arrangement is termed a single-pass heat interchanger.