1. In 1894 Humbert encountered a twisted curve C7, of order 7 and genus 5, the locus of points of contact of tangents from a fixed point N0 to those twisted cubics which pass through five fixed points N1, N2, N3, N4, N5. The cubics of this family which touch an arbitrary plane do so at points on a conic, and it was by investigating this complex of conics that Humbert was led to study C7.

The basic procedures used by most airlines to compile their flight plans are very similar, and are by no means as simple as they could be. When a choice of routes exists it is the normal practice to compile a series of flight plans from which the one giving the most advantageous route in the prevailing weather conditions is selected. In the absence of a direct approach to the problem of selecting the best route, the method of comparing different flight plans is improved by increasing the number of plans; it is therefore desirable that a method of speeding up the process of compilation should be evolved so that a greater number of plans can be prepared. Also the increased aircraft speeds which are to be expected in the future, and the requirement to reduce fuel loads, especially fuel reserves, to a minimum call for some rapid method of modifying the flight plan on receipt of in-flight forecasts and observations.

1. Boltzmann's differentio-integral equation for the molecular velocity distribution function in a perfect gas forms the natural starting-point for a mathematical treatment of the kinetic theory of gases. The classical results of Maxwell and Boltzmann in this theory are well known. They include the proof that, for simple gases, i.e. those in which the molecules have only the three translational degrees of freedom, the only stationary and spatially homogeneous solution is the one which corresponds to the Maxwellian distribution.

The effect on the boundary-layer equations of a weak shock wave of strength ∈ has been investigated, and it is shown that if R is the Reynolds number of the boundary layer, separation occurs when ∈ = o(R−i). The boundary-layer assumptions are then investigated and shown to be consistent. It is inferred that separation will occur if a shock wave meets a boundary and the above condition is satisfied.

The head-on interaction, in the one-dimensional, unsteady isentropic flow of a perfect gas, of a simple compression wave and a simple expansion wave is studied by considering typical examples. The physical aspect of the problem is discussed in (1); in this note the possibility of shock formation is ignored, and the correspondence defined by the complete mathematical solution of the equations of isentropic flow between the x, t-plane and the plane of the characteristic variables is elucidated.

The solution is distinguished by the appearance of two limit lines and a second-order limit point where they meet. It is found that the image of the characteristic plane in the x, t-plane is four-sheeted; all sheets overlap each other, but each covers only part of the plane, and the only point common to all sheets is the second-order limit point, where both limit lines are cusped (§ 3·1).

The solution also contains an edge of regression, and a discussion of the properties of this type of singularity will be found in §§ 2 and 2·1.

In May 1948 the Radio Technical Commission for Aeronautics, a cooperative association of United States government and industrial air telecommunication agencies, issued a report outlining a comprehensive scheme for the development of air traffic control facilities in U.S.A. for the next fifteen years. This report was prepared by a committee set up by the R.T.C.A., Special Committee 31, and it is generally known as the SC31 Report. It was the result of a study undertaken at the request of the Technical Division of the Air Coordinating Committee, an inter-departmental committee established by the Secretaries of State, War, Navy and Commerce, and directed by the President to examine aviation problems of mutual concern and to develop and recommend integrated policies and actions to be taken for their solution.

The use of a coordinate system convected with the moving medium for describing its mechanics, first proposed by Hencky (5), has since been extended by several authors, and has several advantages over the more conventional use of a coordinate system fixed in space; Brillouin(1) has shown that the relation between the strain-energy function for an ideally elastic solid and the stress tensor takes a very simple form when the latter is referred to a convected coordinate system; Oldroyd(8) has given a very general discussion of the formulation of rheological equations of state and has shown that the right invariance properties are most readily recognized when the equations are referred to a convected coordinate system; Green and Zerna (4) have similarly expressed the equations of motion and boundary conditions; and Gleyzal (2), and Green and Shield (3) have applied the formalism to certain problems in elasticity theory.

A theorem of Castelnuovo, which has played a considerable part in the general theory of surfaces, states that any surface which contains a net of elliptic curves is either rational or elliptic scrollar; more precisely, in the first case it is proved that the surface is unirational, and that its unirational representation is obtained by adjoining the irrationality on which depends the determination of one of its points, while the rest of the conclusion follows from Castelnuovo's theorem on the rationality of plane involutions. A somewhat similar result holds for surfaces which contain a net of hyperelliptic curves: thus, it is shown by Castelnuovo (loc. cit.) that, if the characteristic series of the net is not compounded of a g½ the surface is either rational or hyperelliptic scrollar.

The following matrix problems are well known in quantum mechanics:

(a) The one-dimensional harmonic oscillator. Given

determine the eigenvalues hjj of H, and the matrix elements of X, P if H is diagonal. It is found (Wigner (4)) that

Measurements of the areas of magnetization curves of superconducting colloids and films indicate that the difference between the surface energies of the normal and super-conducting states at the same temperature is much smaller than has been suggested hitherto. An attempt is made to explain the high critical fields of thin films, which led to the introduction of the surface energy, by the hypothesis that a negative interphase surface energy in films favours the persistence of superconducting threads in fields higher than those normally regarded as sufficient to destroy superconductivity. A model is proposed of the interface between normal and superconducting phases which allows the interphase surface energy to be positive in pure macroscopic specimens but negative in alloys and thin films and at lattice defects. This enables a qualitative explanation to be given of some of the details, such as supercooling, of the phase transition in pure metals and alloys.