On the flow of electric current in semi-infinite media in which the specific resistance is a function of the depth

In a previous paper the writer has dealt with the flow of electric current from a point-electrode at the surface flowing into a stratified medium each layer of which is of constant specific resistance. In the present paper a general method is developed for determining the distribution of surface potential when the specific resistance is a continuous function of the depth. When this problem has been solved, the surface-potential distribution for any arrangement of discrete or continuously distributed electrodes, which it may be convenient to use in geophysical prospecting by this method, may be obtained by addition or by integration if continuous line-electrodes are employed.

Group characters and algebra

It has been known for some time* that the elements of a matrix of degree n may be arranged in sets which correspond to cycles of the symmetric group of order n !, and that there are relations connecting permanents and determinants, e.g. , /a* p y S a |3 y 8/ ( Further, MACMAHON and BRIOSCHI have pointed out the close analogy which exists between the threefold algebra of the symmetric functions an, hn and sn, and the theory of determinants, permanents, and the cycles of substitutions of the symmetric group. Here we trace the analogy to its source by fixing attention on the characters of the irreducible representations of the symmetric group of linear substitutions, as the centre of the whole theory. By this means divers theories of combinatory analysis and algebra are seen to be merely different aspects of the same theory. For the symmetric group of order n ! the characters are all integers, and we associate with each partition of n both a character of the group and a cycle of substitutions.

The interaction of clay with water and organic liquids as measured by specific volume changes and its relation to the phenomena of crumb formation in soils

The densities of clay particles determined by the specific gravity bottle method vary somewhat according to the liquid used. Thus water, alcohol, and benzene all give slightly different values. The results show that some condensation of the liquid on the surface of the particles usually, if not invariably, takes place, indicating a certain amount of interaction between the clay and the liquid. The investigation described in this paper shows that this interaction depends partly on the exchangeable bases associated with the clay, and partly on the molecular constitution of the liquid. The relations are as follows : (1) No evidence could be obtained of interaction between clay and non-polar liquids. (2) Interaction took place in all the polar liquids examined. Its amount was measured by the reduction in specific volume of the clay as compared with its volume in tetralin, a very convenient non-polar liquid for specific gravity determinations. This reduction in specific volume is approximately proportional to the number of exchangeable cations present, figs. 10, 11, and to the mean density of their surface charge, Table IV.

The solvation of nitrocellulose in acetone-water mixtures

The process of solution is, considering its enormous importance, remarkably little understood. With nitrocellulose as with many other technical products the “ solution ” formed is of a colloidal nature ; that is to say, the ultimate units of dispersion are very large compared with the solvent molecules, and osmotic phenomena play an unimportant part. It is clear that solvation, or attachment of some or all of the solvent to the dispersed particles must be essential to the process of solution, but so far, comparatively little work has been done on solvation probably because of the lack of an unequivocal method for its determination. Yet the uncertainty concerning the solvation of colloidal particles is a source of difficulties in many investigations, to mention only a few, in the determinations of particle-size by settling or centrifuging, in viscosity work, and in light scattering experiments.

The spectra of singly and doubly ionized sulphur in the region λλ 3300-4900 a

The atomic spectra of the elements of the oxygen group in various stages of ionization have been of increasing importance to spectroscopic theory in the last few years. Of no less prominence is their astrophysical importance : in the determination of stellar radial velocities, and in the study of stars of spectral types 0, B, and A, an accurate knowledge of wave-lengths in the high excitation spectra of these atoms is indispensable. For these reasons it has been disappointing to find that the published data on sulphur, in particular, are of inadequate accuracy. The early observations of EDER and VALENTA* and of EXNER and HASCHEK are incomplete and hardly reliable to 0.2 A. Wave-lengths listed by BHATTACHARYYA appear to be in error by amounts up to 0.5 A., whilst Ingram’s measurements in S II§ include only the stronger fines, and in S III|| only those which have been classified. Recent work of comprehensive scope is due chiefly to L. and E. BLOCH and to GILLES,** who unfortunately record many fines ascribable to impurities. In addition, the wave-lengths of their fines are sometimes in disagreement by as much as 0 • 1 A. It therefore seemed desirable to investigate the spectra S II and S III, beginning with the region of greatest astrophysical importance, viz., from X 3300 to X 4900 A., and taking particular care not only to obtain accurate wave-lengths, but also to exclude all impurity fines.

The plastico-viscous deformation of right circular cylinders of soft metal under variable load axially directed

1. Soft metals are of service to man in many ways, extended use following each advance in knowledge. The present investigation had its origin in an attempt to account analytically for the many apparently irreconcilable properties exhibited by right circular cylinders of soft copper when subjected to appreciable strain under heavy crushing loads. During the course of the work it became clear that if advance at all were to be made general methods of analysis would require to be adopted ; and that results forthcoming would be applicable to materials, of a similar nature, other than the one directly used as a standard of comparison. The presentation of the subject matter has accordingly been arranged to give prominence to theoretical and rational aspects, references to experiment being rather for comparison than for support to the argument. Two publications will be found of service in the present connection, the one illustrative of a cogent point in the theory, the other descriptive of practice. They will be referred to as I, and II, respectively.

On the motion of an elliptic cylinder through a vicous fluid

The NAVIER-POISSON equations for the flow of an incompressible viscous fluid are not, as yet, am enable to complete mathematical solution. A number of approximate solutions to them have been obtained in certain special cases, the greater number of these relating to the slow steady motion of a very viscous fluid, i.e., to conditions when the Reynolds’ number is very small. The solution due to STOKES for the flow past a sphere is based on the assumption that the inertia terms in the viscous equations are negligible. A solution for the flow past a cylinder in the presence of walls has been obtained by BAIRSTOW, CAVE and LANG, making the same supposition, also by BERRY and SWAIN for an elliptic cylinder and by FRAZER for a number of conditions, whilst BASSETTS obtained a solution for the flow in the neighbourhood of a sphere moving impulsively from rest.

Determinations of the fundamental standards of length in terms of wave-lengths of light

A new apparatus for determining the relationship between wave-lengths of light and the fundamental standards of length has been previously described.* Definitive determinations have now been completed of the lengths of the yard and metre in terms of the wave-length of the cadmium red radiation, both in air and in vacuum, and the present paper gives the results of these determinations. Previous determinations have been made by MICHELSON and BENOÎT, by BENOÎT, FABRY, and PEROT,} and by WATANABE and IMAIZUMI, of the length of the metre in terms of the cadmium red radiation in air, and these results, after adjustment as nearly as possible from the experimental data available to uniform conditions, agree with each other and with that obtained in the present work, within a total range of four parts in ten millions, a range which is not greater than may reasonably be attributed to the experimental errors of determination of the lengths of the different copies of the metre against which the several comparisons have been made. No previous direct measurement has been made of the length of either the yard or the metre in terms of wave-lengths in vacuum. The paper records the first independent determination of these important relationships, and incidentally affords a new direct determination of the refractive index of dry air, free of carbon dioxide, which is in good agreement with that given by PÉRARD, but differs appreciably from that given by MEGGERS and PETERS.

The freezing points, melting points, and solid solubility limits of the alloys of sliver and copper with the elements of the b sub-groups

It is well known that both copper and silver, when alloyed with many other elements, are able to form primary solid solutions of the substitutional type, in which the solute atoms replace those of the solvent upon its lattice so that the crystal structure of the parent metal is retained. The solubility limits of many of these solid solutions have been determined experimentally, but little progress has previously been made in discovering general principles or a quantitative theory. In the present paper we confine our attention to the alloys of copper and silver with the elements of the B subgroups, including those of the two first short periods.

An extension of Waring's problems

An interesting extension of Waring’s famous problem is the following:— Can every sufficiently large n be expressed, as the sum of s almost equal k-th powers; or, more generally, can every sufficiently large n be expressed as the sum of s positive k-th powers almost proportional to s arbitrarily assigned positive numbers Xl5 x2, ... x, ? I have developed two methods to discuss this problem, one based on the Hardy-Littlewood method for the solution of WARING’s problem and the other on the new VINOGRADOFF method* for the solution of the same problem. In this paper I shall discuss the case k3 by the first of these methods. The case of five or more squares may be treated in the same way, and the results are similar; somewhat deeper and more troublesome analysis is required to deal with the case of four squares. The principle of the method employed here is that of weighting ” the various representations of n as the sum of s k-th powers in such a way as to make predominant the particular representation of which we are in search.