The development during the last two decades of analytical techniques for the solution of problems relating to the optimisation of rocket trajectories is outlined and the present position in this field of research is summarised. It is shown that the determination of optimal trajectories in a general gravitational field can be expressed as a Mayer problem from the calculus of variations. The known solution to such a problem is stated and applied, first to the special case of the launching of an artificial satellite into a circular orbit with minimum expenditure of propellant and, secondly, to the general astronautical problem of the economical transfer of a rocket between two terminals in a gravitational field. The special cases when the field is uniform and when it obeys an inverse square law of attraction to a point are then considered, and the paper concludes with some remarks concerning areas in which further investigations are necessary.

The subject of three-dimensional flow in axial compressors and turbomachines has been extensively studied since 1945. This paper gives a means of calculating the approximate three-dimensional flow in an axial compressor by giving expressions for the slope of the velocity profile as a function of the axial co-ordinate. These expressions bring out the rôle of the ratio of stage inlet annulus height to stage length in the three-dimensional flow in these machines. The effect of the three-dimensional flow on the stage temperature rise at mean radius is discussed by introducing a work done factor.

A description is given of a supersonic pressure-tube wind tunnel which has been constructed at A.R.D.E. This is a blow-down tunnel which uses as a reservoir a long tube filled with gas under pressure. A quasi-steady supersonic flow is achieved by expanding in a convergent-divergent nozzle the subsonic flow behind rarefaction waves which propagate down the tube when a diaphragm at the nozzle exit is burst. The theory of the operation of the tunnel is given and calculations are made of the boundary-layer growth along the tube. Pressure-time records were obtained in the tube, and a high speed camera was used to obtain pictures of the flow round a model. Measurements also included a pitot-tube traverse of the nozzle exit, and the Mach number distribution was determined from the ratio of the pitot to the stagnation pressure. Tests showed that, as predicted, a constant stagnation pressure was obtained ahead of the nozzle, and it is considered that a tunnel of this type would be a cheap and simple way of obtaining an intermittent tunnel with adequate running time for many types of test, and capable of operating at a Reynolds number of more than 107 per inch at a Mach number of about 3·5.

It has been shown in another paper that the relation between the compressive buckling stress coefficient and side ratio for a rectangular plate with the sides elastically restrained against deflection and rotation can be obtained from the corresponding curve for a plate with the same end conditions but with both sides simply-supported, provided only that the buckling stress coefficient and wavelength of the buckle are known for an infinite strip with the given side conditions. This paper presents these numerical values for infinite strips with a wide range of edge stiffnesses.

An understanding of radial flow between confined boundaries is of practical importance in the design of radial diffusers and air bearings. This study presents a combined experimental and theoretical analysis of radial flow, without swirl, between parallel discs using air at incompressible speeds.

Emphasis is placed on the pressure distribution sufficiently far downstream of the channel inlet for the entry conditions to be unimportant. However, a study is also made of the main features of the flow near the inlet, particularly within the annular separation bubble.

It is shown, for both turbulent and laminar flow, that a similarity solution is possible only in special cases where certain terms in the equations of motion can be neglected. Approximate solutions are obtained for the turbulent and the laminar radial pressure distributions using an integral momentum method. Both theories agree well with experiment. The critical Reynolds number for reverse transition is found to be approximately the same as that for flow in twodimensional channels and circular pipes. With the flow separating at the channel inlet, it is established that both a suitably chosen, minimum pressure coefficient of the separation bubble and the reattachment distance are functions only of the channel width for a given inlet pipe diameter and are independent of Reynolds number and the diameter of the discs.

Based on earlier theories of mechanical, electrical and magnetic relaxation phenomena, the paper discusses a simple mathematical model representing qualitatively the basic characteristics of “internal” damping in solid materials. The model is assessed with respect to variation with frequency of stiffness and loss factor at simple harmonic oscillations, and is applied to oscillatory systems with discrete parameters and to continua.