Before the introduction of interpolation formulae, beyond linear interpolation by proportional parts, the presentation of the numerical values of mathematical functions was much restricted, for the labour of computation and the cost of printing, to say nothing of the inconvenience of handling a bulky volume, had to be increased quite disproportionately with every increase in accuracy. A four-figure logarithm table occupies two small pages, Chambers's seven-figure table takes 150 pages, while Vega's ten figure table requires 300 pages twelve inches long.

The circles in a plane can be represented by the points of space of three dimensions in such a way that the points (point circles) of the plane correspond to the points of a quadric Q in space and two orthogonal circles in the plane correspond to two points conjugate with respect to Q. An ∞2 system σ of circles in the plane corresponds to a surface S in space and the reciprocal Sσ′ of circles in the plane closely and reciprocally related to the system σ. So far as I know only trivial examples of such systems σ and σ′ have hitherto been noticed and my object is to direct attention to an illustration drawn from the modern geometry of the triangle which is less simple and perhaps more interesting.

The theorem that if two triangles be inscribed in a conic their six sides touch another conic is, of course, to be found in all the text-books; it is apparently due in the first place to Brianchon. The further remark, that if three triangles be inscribed in a conic the three conics obtained from them in pairs have a common tangent, is to be found in Taylor's Ancient and Modern Geometry of Conics; it was made independently by Wakeford.

The following is an attempt to account theoretically for the observed specific heat of water vapour. The usual theory of the vibrational specific heat together with the theory of the dissociation of the H2O molecule at high temperatures is found to account satisfactorily for the observed facts. It is shown, also, that at temperatures near 100° C. there is an effect due to polymerisation.

Varying ideas of the mechanism of the propagation of light and of the constitution of matter have led at different times to different formulae for the dependence of the refractive index n of a medium on its density ρ, notably those of Laplace, , and of Lorenz and Lorentz, . Recently Macdonald has given a theory which seems more in accord with the modern concept of atomic structure and finds a formula for a non-magnetic medium. It has therefore seemed worth while to examine a number of experimental results to see whether this new formula gives as satisfactory agreement as that of Lorenz and Lorentz; it is found that it does in most cases, and is occasionally the better.

The following theorem, first enunciated by Dr Zeeman in 1899, has attracted the attention of several writers:

If five points A, B, C, D, E in three dimensions are such that the feet of the perpendiculars from E on the faces of the tetrahedron ABCD are coplanar the relation between the five points is symmetrical.

The paper deals with the eddy currents in thin circular cylinders of uniform conducting material, due to periodic currents in conductors lying parallel to the axis of the cylinder, or to the rotation of the cylinder in a two-dimensional field of force. The first of these problems was discussed by Mr M. B. Field in a paper entitled “Eddy current losses in three-phase cable sheaths,” read before the British Association at their Cambridge meeting in 1904. The solution proposed, however, although probably sufficient for the object, is mathematically defective, in that the field due to the current carried by the cable is assumed as the total field, the effect of the eddy-current field on the eddy currents themselves being left out of account.

The early experiments of Lenard(1), Leithauser(2), and Becker(3) on cathode rays led to the exponential law of absorption; those of Rutherford(4), Schmidt(5) and Hahn and Meitner(6) on β-rays indicated that the rays from some radioactive bodies were absorbed exponentially and that those from others were not. It was considered by Hahn and Meitner that the exponential law of absorption was an indication of the homogeneity of the rays. It was found also that the exponential law was not disturbed by the thickness of radiating material and this led to the conclusion that the rays are not appreciably reduced in velocity on their passage through matter. The theories of McClelland(7) and Schmidt(8) attributed the absorption to the stoppage of particles in mid-career.

The method described gives the differential energy distribution curve directly from readings of a galvanometer and a voltmeter. When adjusted, the arrangement is convenient in operation and very good “resolution” can be obtained. The existence of a maximum frequency of secondary electrons at a finite small energy has been established, and its position is shown to be independent of the energy of the primary beam over a fairly wide range. The effect of deposition of tungsten on a clean copper surface is studied with respect to the secondary emission properties of the surface. From these experiments some interesting conclusions concerning the mechanism of the process may be drawn.

I am very grateful to Prof. Sir Ernest Rutherford for the encouragement he has given and the interest he has taken in this work.

The locus of a point the feet of the perpendiculars from which on the faces of a tetrahedron are coplanar is known to be a cubic surface having nodes at the corners: the pedal planes, by which I mean planes containing the feet of four perpendiculars, do not seem to have been much discussed. I propose to prove two theorems concerning them and make some remarks on their envelope.

We recently had occasion to have a right-angled quartz prism made with the optic axis of the quartz perpendicular to one of the short sides of the right-angled triangle (Fig. 1). The prism has the property that it gives two images when light enters perpendicular to one of the faces and, after internal reflexion at the hypotenuse, passes out at right angles to the other face. It was at first sight rather difficult to see why this doubling of the image occurs, since the incidence on both faces of the prism is normal and quartz is a uniaxial crystal, there can be no double refraction occurring.

The Langmuir-Frenkel theory of surface action has been shown to involve a simple relation between the partial pressure of the catalyst poison, and the reduction in reaction velocity which results. Very interesting observations have been made by Beebe showing that the heat of adsorption of carbon monoxide on a reduced copper catalyst falls considerably and reaches a limiting value as the surface approaches saturation. Pease and Stewart have shown that the presence of less than one percent, of the total carbon monoxide that a given copper catalyst can adsorb will reduce the reaction velocity of the combination of hydrogen and ethylene at the surface of the copper by 88 per cent.

Much attention has been focused on the excitation of atomic spectra by electrons whose velocities do not greatly exceed the critical value, and it is now possible to form a rough idea of the behaviour of the “excitation function,” or probability that an electron of given velocity shall excite on impact, when the energy of the electron is only slightly greater than that required for excitation. But the form of this excitation function for much higher velocities is still in considerable doubt.

The motion of a light string loaded with equal masses at regular intervals has been discussed by operational methods. It is found that the system, though possessing only a. finite number of degrees of freedom, shows most of the characteristic features of dispersion. The operator e−σx/c that occurs in the discussion of uniform continuous strings is found to arise as the limit of an operator defined wholly in terms of definite integration, and its interpretation involves a theorem analogous to Taylor's theorem, but apparently more general. There is no phenomenon analogous to the conduction of heat. Any irregularity in the distribution of mass, however, produces strong internal reflexion of the shortest waves, and may provide a mechanism for conduction.

A method of using a valve for amplifying ionisation currents 100,000 times is described, which avoids the instabilities usually associated with such apparatus. The necessary conditions to be satisfied, while extremely simple, are based on theoretical conditions which are discussed in full. Used in conjunction with a galvanometer of sensitivity 200 mm. per micro-ampere, the system behaves in a similar manner to a low capacity quadrant electrometer of sensitivity 6350 mm. per volt, shunted by a leak of 360 megohms, the value of the latter being slightly greater for negative currents than for positive. In addition, it has the advantages of portability, compactness, and ease and rapidity of erection and operation, being specially suitable for lecture demonstrations.

The method of compensation can also be applied to other valve circuits, resulting in a much steadier zero.