Sommerfeld has shown that the electromagnetic field set up by a Hertzian oscillator placed on the infinite plane surface separating two media normal to that surface may be obtained from the Hertzian function of cylindrical coordinates r, z,

where the suffixes 1, 2 refer to the two media occupying the regions z>0, z<0,

and

where ɛ1, μ1, σ1 are the dielectric constant, permeability, and conductivity (in Heaviside units) of the first medium, k2, k2 like expressions for the second, c is the velocity of light, and n the frequency in radians per second; +(λ2−k2)½ is taken to have its real part positive.

In the geometry of the cubic curve Γ in space, we have to study the quadrics Q which stand in certain special relations to the curve. We shall find it convenient to refer to these relations as A, B, B′, C, C′; these are defined below.

The vertical gradient of temperature needed to produce convection currents in a layer of incompressible liquid has already been investigated in several instances. For a compressible fluid instability cannot arise until the gradient exceeds the adiabatic one; it has been assumed usually that what matters is the excess of the actual gradient over the adiabatic. Thus the excess needed in a compressible fluid is to be found by the same formula as gives the gradient needed for instability in an incompressible fluid. It is desirable, however, to investigate the validity of this assumption.

A type of automatically interrupted triode oscillations is described which depends on the interaction between an oscillating triode circuit and. a circuit containing a non-linear resistance and a time-constant device. The theory of the circuit is developed and tested experimentally by means of oscillograms. The theory of the rise and decay of currents in a circuit containing an inductance and a non-linear resistance is dealt with in an appendix.

Consider the series

and let

The series

is not necessarily convergent. Suppose that its Poisson sum exists. This we denote by

The expression

when existent will be said to be the sum (, p) of the series (1).

For the quantitative examination of the behaviour of two-dimensional insoluble films on water surfaces the trough apparatus devised originally by Langmuir has been hitherto employed. With the aid of this apparatus it has been possible to demonstrate the existence of two-dimensional films in the solid, smectic, liquid and vaporous states, to observe the phenomenon of two-dimensional dimorphism, to study the conditions of phase equilibria and to measure the entropy changes associated with the process of two-dimensional fusion, vaporisation and sublimation. No very definite concept of molecular orientation has been obtained from the force area diagrams. Apart from the fact that the polar groups adhere to the surface and that the hydrocarbon chains in long chain compounds adhere to one another, our picture of all forms of films except possibly the solid and the highly attenuated or vaporous states is by no means clear and the various suggestions made as to the molecular architecture are somewhat problematical. Two other methods suggest themselves for gaining a further insight into molecular structure: optical methods on the lines outlined by Rayleigh and by Raman and measurements of the interphasic potential differences. While it may be possible in the future to develop the former into a method of precision, we have succeeded in rendering the latter as accurate as the method of the Langmuir trough. That interphasic potentials between air and liquid exist and are of not inconsiderable magnitude has long been known, although the first systematic investigations on changes in these potentials on the addition of capillary active materials to the liquid phase were made by Kenrick (Z.P.C. xix, 625,1896). The technique was considerably improved by Guyot (C.R. clix, 307, 1914, Ann. d. Phys. x, 2501, 1924) and by Frumkin (Z.P.C. cix, 34, 1924, cxi, 196, 1924, cxv, 485, 1925).

The usual methods of investigation of asymptotic expansions of the various types of Bessel Functions show that the remainder is less in absolute value than the first term neglected. A more refined result was obtained by Stieltjes for K0(x) and certain other functions of order zero; he found an asymptotic series for the remainder and showed that the error due to stopping at one of the smallest terms is of the order of half the first term neglected. Watson notes that it would be of some interest to obtain corresponding results for functions of any order, and this is the object of this note. The method is quite different from that of Stieltjes.

A microcalorimeter is described accurate to 0·0005 cal. The elevation of temperature was measured by a series of iron-constantan thermo-couples, with one set of junctions making good thermal contact with the tube-where the heat was liberated and the others in a brass ring outside, kept in a thermostat. They were connected in series with a very sensitive moving coil galvanometer. The Tian multiple walled thermostat was used—three concentric copper cylinders insulated with kapok, the inner filled with water and the outer controlled by a mercury regulator. The constancy of temperature in the inner vessel was rather better than 1/500,000° C.

Continuous heat evolutions could also be measured by passing a current through another set of thermo-couples, when the Peltier cooling compensated for the heating and the temperature was kept down to its original value, thus avoiding cooling corrections.

When a floating body performs free simple harmonic vibrations in a vertical plane it will create a field of pulsating current flow in the liquid, and the total kinetic energy of the whole moving system will correspond to that of the floating body with an enhanced mass. The total apparent mass of the body is often called the “virtual mass” and the apparent increment the “added mass,” and these terms will be used subsequently. We are concerned in this paper with finding the “added mass” of rectangular and triangular prisms floating in water. The authors' concern with this problem is related to their investigation of the natural frequency of lateral vibration of a ship's hull, and they have contributed already two papers to these Proceedings which should be associated with the present one. In their ultimate problem the body is not moving as a whole in a vertical plane but is flexing about two or more nodal points, but in the present problem the floating body is supposed rigid and all parts of it are moving vertically with the same velocity. It is believed the present investigations are applicable to the lateral vibrations of ships' hulls, but whether or not this is so the investigation is complete in itself: for example it may be used to calculate the natural frequency of a punt or pontoon which has, say, been relieved suddenly of a dead load.

The formulae considered in this paper deal with the multiple tangent lines and planes to surfaces in three dimensions; the functional method employed is that of a previous paper, in which a more restricted discussion was given for surfaces in higher space.

If a light quantum has an electric moment, we should expect its axis to coincide with the direction of the electric vector, and therefore to be perpendicular to the plane of polarisation. Now an electric doublet of moment μ on entering a uniform electric field E with its doubletaxis parallel or antiparallel to the field will have its energy changed by an amount ∓ μE. For a light quantum this is equivalent to a change in wave-length

when in the field. The method adopted to detect this change was to set up a grating with a strong electric field normal to its surface, and then have a plane polarised parallel beam of light fall on the ruled surface with the electric vector parallel to the direction of the field. The spectra formed near the normal to the grating were then examined both in the presence and absence of the field.

1. We consider a small cube of the solid, of edge l, in the actual strained position. If the stresses were removed and the cube cooled to the absolute zero, the displacements of its particles would be (−u, −v, −w). Then the particles may be considered to have displacements (u, v, w) from a standard position; and the cube has strain components of the usual forms

The expansion of a general function in a series of squares of Bessel functions has been considered by Neumann. As, however, there appear to be only a fairly small number of known expansions of this type, it is perhaps worth while adding to their number. I give here some new expansions of this type, and deduce some simple inequalities satisfied by Bessel coefficients of low order.

The equation of state of an ionised gas has been discussed from time to time by various writers*. It is of importance chiefly in astrophysical applications. The simplest form of the problem, and the only form considered in this note, is to find the pressure in a gas of given temperature in which the numbers of atoms per unit volume at the different possible stages of ionisation are supposed known. In describing the ionisation an electron is counted as “free” when its total energy is positive. If this problem can be solved we should have a good estimate of the relative importance of the various physical factors at work, without tackling the more difficult, though more natural, problem of finding the ionisation for given temperature and pressure.

Mr Ursell has given a proof in these Proceedings (vol. 24, p. 445, (1928)), that there exist non-combining groups of wave functions of an assembly of similar systems. His proof was not sufficient; but it can be amended so as to be rigorous. It will be necessary to suppose, however, that the wave function ψt is analytic for all real values of the time, for each set of space coordinates: q1, q2, q3, q4….

The planes of a given S5 can be represented by the points of a locus V9 in space of nineteen dimensions. This locus is a double locus on a certain other manifold V14, and the tangent spaces of the V9 generate a W18. Segre has a memoir on the subject of these loci, in which he arrives at his results by a series of short steps, the argument being mainly geometrical.

The boundary layer equations for a steady two-dimensional motion are solved for any given initial velocity distribution (distribution along a normal to the boundary wall, downstream of which the motion is to be calculated). This initial velocity distribution is assumed expressible as a polynomial in the distance from the wall. Three cases are considered: first, when in the initial distribution the velocity vanishes at the wall but its gradient along the normal does not; second, when the velocity in the initial distribution does not vanish at the wall; and third, when both the velocity and its normal gradient vanish at the wall (as at a point where the forward flow separates from the boundary). The solution is found as a power series in some fractional power of the distance along the wall, whose coefficients are functions of the distance from the wall to be found from ordinary differential equations. Some progress is made in the numerical calculation of these coefficients, especially in the first case. The main object was to find means for a step-by-step calculation of the velocity field in a boundary layer, and it is thought that such a procedure may possibly be successful even if laborious.

The same mathematical method is used to calculate the flow behind a flat plate along a stream. The results are shown in Figures 1 and 2, drawn from Tables III and IV.