A three-filament mechanism is described which will enable multimolecular films to be studied by the method of contact potentials and this method is applied to the study of the evaporation of sodium films of various thickness from one to twenty layers. It is shown that the vapour pressure and heat of evaporation both increase as the thickness increases from one to three layers and thereafter become practically independent of any further increase in the film thickness. The vapour pressure for a thickness of four layers is within experimental error the same as that for bulk sodium.

A difficulty due to the effect of the evaporating stream of atoms annulling the space charge around the emitter filament is discussed, and attempts are made to use this effect to measure the rate of evaporation at temperatures from 500° K. to 825°K.

13. Formulae have been obtained for the moment of the fluid forces which act on a plane plate which is gliding on the surface of a stream of finite or infinite depth. It has been shown that the formulae for the lift and moment forces acting on a plate which is gliding on a stream of infinite depth may be obtained by a limiting process from the formulae for the general case of gliding on a stream of finite depth. Also, by another limiting process, the lift and moment forces for the Rayleigh flow past a flat plate in an infinite stream have been obtained from the lift and moment forces for gliding on a stream of infinite depth.

Numerical work has been carried out for the case when the angle of incidence of the plate to the stream is 5° and this has shown the variation of the lift and moment forces and the position of the centre of pressure with a range of values of the length of the plate, the depth of the stream and the height of the trailing edge of the plate above the bed of the stream.

The technical importance of constructing valve amplifiers disturbed by as little background noise as possible has led to a very searching investigation of the causes of this defect. Some causes—bad contacts, changes in the surface of the filament, secondary ions from the grid, and so forth—though technically speaking not trivial, are of minor physical interest; but it has long been recognized that there are two which cannot be dismissed so easily.

An elementary theory of the diffraction of light by supersonic waves was put forward by Sir C. V. Raman and myself (4) to explain many of the important features of the phenomenon observed by Bär and others. Recently Mueller (2) pointed out the essential idea that the optical diffraction effects in solids may be interpreted on the basis of this theory if due account is taken of the photoelastic effects arising from the periodic strains caused by the sound waves. He was thus able to give an explanation of the results of Schaefer and Bergmann (5, 6). Mueller's idea, however, required a still further refinement, since it did not consider the phase relationships between the differently polarized components of the diffraction orders. This refinement has, however, been recently incorporated by Mueller and myself (3). This led us to offer an explanation of the diffraction effects observed by Hiedemann and Hoesch (1) in glass blocks.

1. Let f(x) be a real function of period 2π, integrable L over (0, 2π), and let

By sn(x) and σn (x) we denote respectively the partial sums and the first arithmetic means of the series (1·1). Similarly, by and we denote the partial sums and the first arithmetic means of the series

conjugate to (1·1). By we mean the function conjugate to f(x), that is

where the integral is taken in the principal-value sense.

By measuring the resistance of a phosphor bronze wire in thermal equilibrium via various substances with crystals of iron ammonium alum it is shown that the time for thermal equilibrium between the ionic magnets of the salt and its lattice vibrations is less than 0·5 sec. for all temperatures above 0·025° T*. When liquid helium or a german silver tube forms part of the cooled portion of the apparatus, the time for equilibrium is increased to a few seconds for temperatures below 0·4° K. It appears possible that the thermal conductivity of german silver is less than 10−8 cal. cm.−1 sec.−1 degree−1 below 0·05° T*, and there are indications that the thermal conductivity of liquid helium at temperatures below 0·3° K. is small compared with its value at 2° K.

No ionization was found in hydrogen leaving palladium, either when the gas diffused through the metal or was evolved by warming palladium previously charged with hydrogen by electrolysis in dilute sulphuric acid.

A Cremona transformation Tn, n′ between two three-dimensional spaces is said to be monoidal if the surfaces of order n in one space which form the homaloidal system corresponding to the planes of the second space have a fixed (n − 1)-ple point O. If the surfaces of order n′ forming the homaloidal system in the second space have a fixed (n′ − 1)-ple point O′, the transformation is said to be bimonoidal. A particularly simple bimonoidal transformation is that which transforms lines through O into lines through O′, and planes through O into planes through O′. Such a transformation we shall call an M-transformation. Its equations can, by suitable choice of coordinates, be expressed in the form

where φn−1(x, y, z, w) = 0, φn(x, y, z, w) = 0 are monoids with vertex (0, 0, 0, 1).

It is shown that the soft X-ray emission bands from Be and A1, present in small quantities in solid solution in Cu and A1, have a form without a sharp highenergy edge, the feature most characteristic of these bands in pure metals. The diffuse form of the bands indicates (1) that all the valence electrons of the impurity atom are given up to the lattice, (2) that the impurity atom refuses to take within it those electrons of the lattice which have energies close to the maximum energy of the conduction electrons.

The notion of fiducial probability was introduced by R. A. Fisher and is now widely used in statistical work involving a single unknown parameter. Fisher has also considered the joint fiducial distribution of several parameters for which there exists a sufficient set of statistics, and has derived the fiducial distribution of the two parameters of the one-variate normal law*. Since Fisher's account is brief, we give a more extended description of the fiducial distribution of several independent parameters possessing a sufficient set of statistics.

The conductivity of thin films of the alkali metals has recently been measured in the H. W. Wills Physical Laboratory, Bristol*. It was found that as the thickness of the film is decreased to that of a few atomic layers the conductivity drops below that of the bulk metal. In the papers quoted the hypothesis was put forward that this effect is due to the shortening of the mean free paths of the conduction electrons of the metal by collisions with the boundaries of the film. The experimental results were compared with a formula derived on the basis of this hypothesis. This formula was, however, obtained subject to a number of simplifying assumptions, and it is the first purpose of this paper to obtain a more accurate formula. I also compare this formula with experiment, and make certain deductions about the surfaces of thin films.

By using Bragg and Williams' approximations and Bethe's interaction energies between nearest neighbours, it has been shown that, for alloys of the AB3 type, (1) the maximum value of the critical temperature Tc must occur when c = ½ (2) for all temperatures within the range 0 < T < (Tc)max there is always a concentration range within which the alloy exists in a two-phase state, the one phase being partially ordered and the other disordered. This two-phase region is small at all temperatures and vanishes both at T = 0 and at T = (Tc)max. In particular for T/(Tc)max = 0·82, which is the value of Tc for c = ¼, the limits of the two-phase region are c = 0·245 and c = 0·256. The theoretical phase diagram has therefore qualitatively the shape, given in Fig. 5. For a fixed concentration there is thus a certain temperature range in which a disordered phase and a partially ordered phase exist in equilibrium with one another. The temperature at which the disordered phase completely disappears is that usually given by experimental determinations of the critical temperature. It is somewhat higher than Tc, but its maximum value also occurs when c = ½, so that the discrepancy between theory and experiment for this type of superlattice still remains.

1. Particular series of conal and toroidal functions have been studied by various authors*, but so far no method has been given of expanding an arbitrary function in such a series. Ordinary methods fail, since conal functions (and also toroidal functions) are not orthogonal, although they form a complete set. In this paper I give an expansion theorem.

This paper may be regarded as a sequel to a previous papers(1) in these Proceedings. The vector and matrix notation of that paper used for a statistical sample is systematized somewhat further, so that while a sample S refers as before to the matrix of nm values (a sample of m observations in one variate only being a row vector), we write

for the linear regression formula between the dependent and independent variates into which a sample is supposed partitioned (in place of equation (12) of (1)). More generally, a third submatrix S0 is partitioned off, and its effect eliminated (corresponding to equation (13) of (1)), but without loss of generality we assume that S2 in equation (1) above can always stand for S2.0 if necessary.

A ternary form of degree n can be expressed as a symmetrical determinant, of n rows and columns, whose elements are linear forms; furthermore, not only is such a mode of expression known to be possible, but A. C. Dixon, in 1902, gave* a process by which the determinant can be obtained when the ternary form is given. This process, however, although it admits of such a straightforward theoretical description, cannot be carried through in practice, for a general ternary form, without the introduction of complicated algebraical irrationalities, even if we restrict ourselves to forms of the fourth degree; consequently no application of Dixon's process to an actual example seems to have been published. If then a choice can be made of a quartic form for which the reduction to a symmetrical determinant can be carried out without undue complication, it seems fitting to give some account of it. The following pages are therefore devoted to the study, from this aspect, of the form x4+y4+z4, for which the reduction can be accomplished without introducing any irrationality other than the fourth root of − 1.

The present discussion of the conduction levels of copper and silver was undertaken in view of the experimental investigations of the absorption and emission bands of metals, in the soft X-ray region, which have been carried out in this laboratory and elsewhere*. It is a well-known result of the electron theory of metals that, for cubic crystals, of lattice constant a, if k is the wave vector of an electron moving in the crystal lattice, there are discontinuities in the energy E of the electron for values of k such that

(n1, n2, n3) being the Miller indices of the planes for which Bragg reflexion of the electron waves takes place. If we draw the energy E as a function of k for a particular direction in the lattice, then we obtain, using the usual notation, curves such as those shown in fig. 1, d being the half-distance between the corresponding planes for which Bragg reflexion takes place.

It is familiar that if on an algebraic surface there is an exceptional curve, that is an irreducible rational curve of virtual grade − 1 when no points of it are assigned as base points, and if there is on the surface a canonical system containing some actual curves, so that pg ≥ 1, then the exceptional curve is a fixed constituent of every curve of the canonical system, generally a simple constituent, and in that case has no intersections with the residual constituent. More generally, if there is on the surface a reducible exceptional curve, i.e. a set of curves which can be transformed into the neighbourhoods of a family of simple points (some of which are in the neighbourhoods of others) on a surface birationally equivalent to the given one, then the canonical system has as a fixed constituent of all its curves at least that combination of the curves which corresponds to the sum of the total neighbourhoods of the points, and generally just this combination, in which case this fixed part has no intersection with the residual variable part.

The absorption limits for the primary β-particles of mesothorium 2 and uranium X2 have been determined using modified tube counters of various types. Results in good agreement with previous single determinations with gold-leaf electroscopes, and at variance with deductions from expansion chamber observations, show the need for caution in fixing limiting energies in β disintegrations from measurements on a relatively small number of tracks.