The eigenvalue equations for symmetric vibration of a complete aircraft are derived in a very general form. The “lumped mass” approximation to the continuous mass distribution is used and sub-matrices are associated with properties of relatively simple branches of the system. The final eigenvalue equations are expressed in terms of these sub-matrices, so that in a numerical application the physical system, as such, is considered only in relation to the properties of the simple branches. It is assumed initially that the aircraft wing and tail have flexural axes of the conventional type, but it is shown in the Appendix that under certain conditions the treatment can be generalised to cover swept and cranked wing aircraft.

The flow past an airscrew rotating with uniform angular velocity in a uniform supersonic stream is considered from the point of view of axes fixed in the airscrew, the motion relative to these axes being steady. A linearised equation and boundary conditions are found for the potential of the disturbance flow caused by airscrews which produce only small perturbations in the main stream. This equation is solved by expanding the potential in powers of the ratio, χ, of the tip speed of the airscrew to the speed of sound in the undisturbed stream. Equations and boundary conditions are found for the coefficients of each term of the series and it is seen that the term independent of χ satisfies the ordinary linearised potential equation for fixed axes. By subtracting suitable special integrals, the more complicated equations for the coefficients of the various powers of χ can be reduced to this ordinary potential equation. Hadamard's methods have been applied by Puckett, Evvard and, finally, Ward to the problem of finding the flow past a thin wing fixed in a supersonic stream and their methods are applied to find the terms of the series in our case. The first two terms are found explicitly for a particular type of windmill. Approximate expressions are then developed for the torque and the drag on such a windmill and the results applied in special cases. The best working conditions for windmills of this type are found by considering the effect on efficiency and power of varying the shape and the speed of rotation, within the limits imposed by the strength of the material of the airscrew.

Wild has extended to three dimensions the von Kármán-Pohlhausen method for the laminar boundary layer flow over a fixed obstacle and used the method for an infinite yawed elliptic cylinder in a stream. In this paper the method is tested in two ways (which may be called full-Pohlhausen and semi-Pohlhausen) for the case of an infinite yawed cylinder when the velocity outside the boundary layer over the surface normal to the generators is of the form U=cxm. The exact solution is known in this case.

A table of the skin friction, displacement thickness and momentum thickness is given for various values of β [=2m/(m + 1)], and the agreement is found to be fairly good for β>0 (accelerated flow) but not so good for β<0 (retarded flow).)

The method developed by G. N. Ward for the treatment of slender pointed bodies in a uniform supersonic stream is applied to three special cases.

• (i) Supersonic flow past a body of revolution with thin wings of symmetrical section and of small aspect ratio at zero incidence.

• (ii) Supersonic flow past a body of revolution with plane wings of small aspect ratio set at incidence to the body, the whole being at incidence to the stream.

• (iii) Supersonic flow past a body of revolution with a plane fin of small aspect ratio set at incidence, the whole being at incidence to the stream.

The pressure distribution on the wing has been calculated for a special case of (i) and is given in the Appendix.