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.

Continued fractions were generalised to more than one dimension by Jacobi and others: later Perron gave an account of the existing state of the subject with a detailed discussion of periodic fractions. Quite recently the subject has been attacked afresh by Mr. Maunsell

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.

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).

The Fourier series of the function defined by

is

and if

then

while

The expression (1) exceeds (2); this is the Gibbs phenomenon.

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.

For r>-1, let denote

If

we say that the series a0 + a1 + a2 +…+an+… is summable by Cesàro mean of order r, or more shortly summable (C, r) to sum s. If r >−1, and

we say that the series is summable by Rieszian mean of order r to the sum s. It has been shown that these two methods of summation are equivalent. Throughout this paper I shall deal with the Rieszian mean, but I shall retain the symbol (C, r). It is known† that if a series is summable (C, r), it is also summable (C, r′) to the same sum for all numbers r′ greater than r.

When a vessel of liquid has been emptied and put aside, a thin film of liquid clings to the inside and gradually drains down to the bottom under the action of gravity. The layer being thin, the motion is very nearly laminar flow, and the curvature of the surface in a horizontal direction may be ignored. Thus the problem for a cylindrical vessel is reducible to that of a wet plate standing vertically.

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.

It is generally believed that isolation has played an important part in evolution. If an organism is to evolve so as to adapt itself to a special type of environment, e.g. a cave or a desert, it must not be swamped in each generation by migrants from the original habitat.

The intensity of stifling is estimated. Under ideal conditions the absorption should vary as the secant of the angle of incidence of the sound.

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.

Many measurements have been made of the complete absorption curves for β-rays in various substances, and it has been found that, for homogeneous β-rays, the curve is not exponential, and, in particular, when aluminium is the absorber, the curve is linear over a considerable portion of its range (e.g. Schonland, etc.).

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.

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).

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.