A modification of the MO method is described which leads to the same polynomial secular equation as does the LCAO method, but via a simpler secular determinant. The basic idea is to divide the whole molecule into parts, calculate the MO's for these by the usual method, and then use linear combinations of these MO's as trial eigen-functions in a variational treatment of the whole molecule.

Equations are developed for a number of cases, and it is shown that the computations of energy levels and MO's from them is often much simpler than from the corresponding LCAO equations.

The object of the paper was to find the forces acting in bubble rafts and to use them for a description of the mechanical properties of these rafts. The first section is based entirely on Nicolson's work and gives expressions for the cohesive forces in the lattice. The second and third sections deal with the repulsive forces arising when a bubble is compressed uniaxially, and are really the basis of the paper. The fourth section introduces the idea of ‘interaction’ between compressions in the three close-packed directions—a conception which explains the breakdown of Cauchy's relations. The conditions of equilibrium in the lattice are investigated in § 5, and in § 6 all these forces are combined to give a complete account of the elastic properties of the rafts.

The remaining section gives a brief account of the experimental checks of the theory; it appears to be a reasonable approximation for small bubbles, and, indeed, we could certainly not expect it to apply to large ones, since for them not only are all the calculations of the attractive potentials invalid because of the large local slopes of the surface, but also the ‘flats’ and ‘caps’ become so large as to interfere with one another.

The forces are considered well enough supported to justify their further use in an investigation of the plastic behaviour of the bubble rafts—an investigation which has explained several interesting experimental facts but brought us little nearer an understanding of the low yield strains of metallic single crystals (see (2)).

In a paper under the above title (Journal, Vol. 1, p. 266) J. P. G. Worlledge has made the useful suggestion that sextant certificates should give a critical table for the corrections instead of the present practice of giving the corrections at a few equally spaced readings. Unfortunately the method which he gives of deriving the proposed form from the present form is unsound. The centring error is correctly stated as

where Z is the sextant reading (the difference between Z and the true angle in this formula is negligible). But part of the centring error is included in the index error, so that to obtain the part of the centring error shown on the certificate the value of C for Z= 0 must be subtracted. This yields

for the formula of the correction.

For the purpose of comparing results with the two wavelengths, 3 cm. and 10 cm., generally used in navigational radar, the Defence Forces Research Institute, in collaboration with the Swedish Admiralty, the Royal Pilot Service and the Marine Cartographical Institute, carried out systematic researches and experiments under various conditions, in view of the special meteorological and navigational circumstances prevailing in Swedish waters. Two radar equipments covering these wave-bands were obtained from the Research Institute and installed on board the marine survey ship Gustav Av Klint during the summer of 1947. A separate report has been issued on the results obtained during the summer months.

If astronomical navigation is only required to give a ten-mile accuracy, as Mr. Parker has suggested in his paper, then its use will be mainly that of a last-aid in emergency. Radio, however, will give this degree of accuracy and, since a radio operator will in any case be carried, will naturally be more popular. On the other hand if astronomical navigation can be made to give a ½-mile accuracy it will become immensely more valuable than any other long-range aid. In the first place the signal received from the stars is completely reliable (at the heights which concern the air navigator, variable refraction can be ruled out) ; it is wholly independent of the ground; it provides a universal coverage which is free and is controlled by a non-political Body; and it can provide an accuracy greater than that of the long-range aids except the limited-purpose systems such as Oboe.

This paper first gives a brief account of the deviation theory applicable to magnetic compasses and a unified theoretical treatment of the principal heeling-error and pitching-error effects. Consequences of symmetry in soft-iron correctors are then considered, and a system of six equal soft-iron spheres at equal distances along three mutually perpendicular axes is shown to have no effect at the compass position, any diametral pair thus having a complete negative counterpart in the remaining four.

This leads to the deduction of the possibility of a simple form of corrector system in which soft-iron correctors remove those parts of the principal inclination errors which are due to induced magnetism. If the remaining part due to permanent magnetism is assumed to be removed by vertical magnets in the usual manner, the correctors when once adjusted would then require no readjustment on change of locality, in distinction from all known correctors at present in use or in course of manufacture, which require continual readjustment in these conditions.

The ‘phase-space’ method in quantum theory is used to derive exact expressions for the transition probabilities of a perturbed oscillator. Comparison with the approximate results obtained by perturbation methods shows that the latter must be multiplied by an exponential factor exp (− ∊/ℏω), where ∊ is the non-fluctuating part of the work done by the perturbing forces; as long as ∊ is small, exp (− ∊/ℏω) ˜ 1 and only dipole transitions have an appreciable probability. As the perturbation energy increases, however, this is no longer true, and multipole transitions become progressively more probable, the most probable ones being those for which the change in energy is approximately equal to the work done by the perturbing forces.

In many compressible fluid flow problems the classical solution breaks down completely owing to the formation of regions of infinite acceleration in the flow field. The actual behaviour of the fluid in such cases does not seem to have been investigated mathematically and this is largely due to the difficulties which enter with non-uniform shocks. It is with these difficulties that this paper is principally concerned.

The way in which discontinuities may arise mathematically in a flow field is first discussed. The equations governing the one-dimensional motion of a gas due to an accelerating piston are then set up. It is shown that when allowance is made for varying entropy conditions due to the presence of non-uniform shocks these differential equations reduce (outside the shock wave) to three first order quasilinear ones.

The initial solution breaks down when a point of infinite acceleration occurs in the flow field. From this point onwards a shock wave grows in the fluid and behind it three different sets of characteristics are required to describe the flow. By working in the plane of two quantities that are constant along two different sets of characteristics, we can use the shock-jump conditions to determine the equation of the shock-line in this plane and to reduce the equations of the characteristics to three differential equations, which would be linear if the relation between the entropy and other flow variables were known.

In the case of a constantly accelerating piston a first approximation is found by neglecting reflexions and entropy variations behind the shock. Using this as a basis we then find a second approximation for the entropy in the neighbourhood of the initial portion of the shock-line and show that the problem reduces to the solution of a second order linear partial differential equation. The introduction of a Riemann function and the satisfaction of the boundary conditions at the shock lead to an integral equation whose solution enables us to determine the position of the shock as a function of the time. The solution is in the form of a power series and is valid provided the shock wave does not become too strong.

Finally, it is shown that if the piston is given a constant terminal velocity a reflected wave from the shock is reflected again from the piston and eventually overtakes the shock and reduces its velocity to a final steady value which is in agreement with the value arising from an impulsive start.

The method proposed consists in measuring by means of a specially designed sextant the altitude and azimuth of a star simultaneously and using these two quantities with the declination to solve the astronomical spherical triangle. In this way, both the latitude and, by means of certain simple additions and subtractions in terms of the local hour angle which is found directly, the longitude are determined.