The object of this paper is to give a method of finding the units of a set of algebraical numbers that shall be convenient practically. Little attention is given to theoretical considerations. The case of numbers composed with the cube root of an integer is treated in § 2, and a few theoretical remarks are added in § 3. In § 4 we show how to test whether the unit found is a fundamental one or not. In § 5 the method is extended to other cases.

The following derivation of the equations of viscous motion, and, by analogy, the equations of motion of an isotropic elastic solid, appears to have an advantage in brevity over those in general use.

The solution for an infinite cylinder filled with viscous fluid, rotating about its axis under the action of a constant couple, has already been obtained by the present writer by the operational method of Heaviside and Bromwich, and previously by other means by T. H. Havelock.

The problem was suggested by Professor G. I. Taylor as being of interest and importance in the phenomena of rupture in a solid, due to the presence of a crack, and the slip in a crystal. In experiments on the distortion of a crystal of aluminium under a tensile stress the conclusion is reached that “…as far as these experiments go, the distortion of a crystal of aluminium under compression is of the same nature as the distortion which occurs when a uniform single-crystal bar is stretched. The distortion is due to slipping parallel to a certain crystal plane and in a certain crystallographic direction, and the choice of which of twelve possible crystallographically similar types of slipping actually occurs depends only on the components of shear stress in the material and not at all on whether the stress normal to the slip plane is a pressure or a tension.”

In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.

It is well known that eclipse observations show that the chief constituents of the higher levels of the sun's chromosphere are ionised calcium and hydrogen. The equilibrium of the calcium has formed the subject of Milne's important researches. The hydrogen presents a rather different problem owing to the large number of possible quantum states that have to be considered.

Where the solutions which carried the lead vein minerals pass along joint planes through the Whin Sill, the dolerite is altered to a white, compact rock for a distance of several feet. The original texture is well preserved in the altered dolerite; the ferromagnesium minerals are replaced by calcium, magnesium and iron carbonate, the feldspars by a minutely crystalline mixture of kaolin and muscovite and the ilmenite by small rutile crystals. The former areas of interstitial quartz are increased and apatite remains completely unchanged. The microscopic evidence that there is neither gain nor loss of alumina during the metasomatism, taken in conjunction with analyses of altered and unaltered rock, may be used to prove that there was a contraction of about 14% during metasomatism. This result is corroborated by the occurrence of cracks which are formed by shrinkage during metasomatism and which are now filled with carbonates.

R. A. Millikan and C. F. Eyring have shown that the auto-electronic emission is independent of the temperature of the cathode up to 1000° K. At higher temperatures a thermionic emission takes place and this masks the auto-emission. W. S. Pforte and the author have recently shown that the thermionic emission is increased by the field in accordance with a relation derived by Schottky, so that when both thermionic and auto-electronic emissions are present it is only necessary to subtract the thermionic current, as given by the Schottky relation, from the total measured current to obtain the value of the auto-emission at the temperature considered. In this way the thermionic and auto-emissions can be separated.

Equations have been given by Moullin to determine the current in a receiving antenna in wireless telegraphy and they have been discussed and extended by Colebrooke, who in particular considered the reception of a plane polarised wave with its electric vector vertical by an antenna which is partly vertical and partly horizontal. His result shows that the intensity of the received signal is independent of the orientation of the receiver. In practice, however, the incident wave front is not vertical, its tilt depending on the wave length and on the resistance of the earth, as shown by Zenneck, and the purpose of this paper is to investigate the effect of the orientation of a bent antenna in the reception of this type of wave. An approximate theory of the Beverage antenna is obtained by the same method.

The heavy elastic structure of a ship possesses many natural periods of lateral vibration, and if any periodic disturbing force within the hull happens to synchronise approximately with any of these natural periods, considerable vibration may result. Experience has shown that the damping of the hull structure is small, and when the rate of revolution of the propeller coincides with a natural period of the ship, a vibration of large amplitude is set up. No doubt reciprocating machinery is competent to produce more violent vibration than turbine machinery, but the inevitable lack of balance of the propeller itself is sufficient to vibrate seriously a turbine-driven ship.

Experiments have been performed to measure the scattering of alpha particles by surfaces on which a source of radioactive material was deposited and on surfaces in the neighbourhood of the source. It was hoped that, where a very narrow but not necessarily homogeneous beam of alpha particles was required, the effective strength of a source could be increased by including particles which are scattered out of the activated surface at glancing angles. The effect was found to be too small to be of any great practical importance.

Experiments were also performed on the scattering of alpha particles by the walls of a long glass tube down which a beam of particles was projected. This is essentially a repetition of some experiments by Lawson and Hess in 1918, whose results indicated that the particles in being scattered by the walls of the tube lost definite amounts of energy. The present experiments indicate that they can lose any fraction of their total energy.

The experiments indicate that the most probable angle of scattering in such experiments is large, and that the average particle spends at least half and probably a greater fraction of its range in the scattering material.

Particular cases of the solution of the equation of tidal-wave propagation in canals of varying section have been obtained by various writers. It is believed that the cases which we are going to consider now have not been investigated before.

If A0, A1, A2, A3 are four arbitrary circles, it is well known that there are eight circles which cut them all at equal angles, and that these eight circles fall into two tetrads B0, B1, B2, B3, C0, C1, C2, C3 which are in desmic position to each other and to the tetrad of circles orthogonal to threes of A0, A1, A2, A3.

It is a familiar fact that an important part is played in the Analytic Theory of Numbers by Fourier series. There are, for example, applications to Gauss' sums, to the zeta functions, to lattice point problems, and to formulae for the class number of quadratic fields.