The necessary conditions of Clebsch and Weierstrass and of the multiplier rule in the calculus of variations, which arise from the study of the first variation of a function, are summarised. A further necessary condition associated with the second variation is stated. The latter condition is applied to two problems: (i) the determination of the thrust-time programme which maximises the altitude of a sounding rocket, (ii) the determination of the thrust direction programme for a rocket with a known propellant expenditure programme which yields a maximum range. In both problems it is found that the additional necessary condition is satisfied.

This paper deals with an unsteady-state three-dimensional heat conduction problem in a prismatic body which has a complicated boundary shape. Numerical techniques such as a relaxation method must be used to solve a problem of this type; this method, however, requires unusually large amounts of labour. The method used in this paper consists of the application of conformal mapping and numerical solution of a partial differential equation and is applied first to a prismatic body of square cross section for which the exact solution is well known. As a second application, the approximate solution for a problem involving a star-shaped cross section is presented.

An analytical discussion on the effect of linear damping on flutter speed of systems having three or more degrees of freedom is described. The study is based on energy considerations introduced in a previous paper, where binary systems were discussed.

A general solution is obtained by using potential flow theory for the design of two-dimensional suction slots for parallel channels. These slots are formed by free streamlines and are so arranged that the channel walls are free from adverse pressure gradient. Where suction slots are intended for the control of boundary layers, the present solution can only be used as a basis for experiment. It is interesting to note that the results also suggest a new method for the design of diffusers by suction.

At flight speeds above about M=2 the cross-sectional area of the fully-expanded jet nozzle tends to be larger than the intake streamtube. The principle of integrating this increase in area of the propulsion unit with the usable volume in order to reduce the after-body drag of the combination is discussed and calculations are made to establish the order of the effect. It is deduced that successful integration of this type can produce worthwhile increases in lift/drag ratio.

The repetitive pattern of buckling, common to most theoretical analyses, is shown to be itself unstable. An approximate but very simple analysis is presented to describe the true buckling behaviour.

Numerical solutions are obtained for the one-dimensional radial flow of a compressible fluid between parallel discs. A general discussion of the variations of pressure, density, velocity and temperature along the radius of the disc is then given. The present investigation covers both subsonic and supersonic flows, a range not previously covered in the literature.

The heat of ablation of glassy materials can be improved by surface-active agents. The degree of improvement is more pronounced at high flight velocities and depends strongly on the coefficient of diffusion in the liquid layer. The analysis is carried out near the stagnation point of blunt bodies for steady state ablation.

The growth of a line vortex with time and the spread of a trailing vortex behind a wing due to turbulence are considered. It is shown that the eddy viscosity for this type of motion may be taken to be proportional to the circulation round the vortex and the solution is then similar to the solution for the growth of a vortex in laminar flow. The method is applied to calculate the distance behind a wing for which the trailing vortices will touch one another.