The average orthodox mathematician who writes or reads books about hydrodynamics is so keen about getting a peg on which he can hang what he calls examples that he quite forgets to apply his formulae to compressible fluids. It appeared to me on reading Colonel De Villamil's letter that a useful purpose would be served by writing the present note; it contains nothing but what anyone could write out for himself, but it was only recently that I made the attempt myself and I should be surprised if many readers have seen the same thing before. We know that in the irrotational motion of an incompressible fluid the velocity potential satisfies Laplace's equation, the present object is to find the corresponding equation for a compressible fluid, to grind it out at full length, and (what is more important than grinding it out) to interpret the result.

In March of last year I had the privilege of reading a paper on the airscrew before the Society, “The Screw Propeller in Air.” The object of this paper was to extend what is usually known as the aerofoil theory of the airscrew, so as to include to some extent the phenomena of slip essential to propulsion in a fluid. This method has since, I believe, been fairly widely adopted both in experimental research work on airscrews and for purposes of designing and analysis.

The present contribution is the outcome of an attempt to modify the previous theory outlined in my paper, “The Screw Propeller in Air,” which theory cannot be considered as being complete although at the time it was written nothing of a similar theoretical nature had been published with the exception of Mr. Lanchester's paper on “The Screw Propeller,” this work being of a purely theoretical nature throughout and not directly based on model aerofoil data obtained from the wind channel.

The previous method of estimating the quantity of fluid, air in the case under consideration, dealt with by the propeller was to assume some constant fraction f say of each annular ring element along the blade, the limiting possible value of f being taken to be unity.

The only experiments on the resistance to torsion of woods that I have been able to find are those of Benjamin Bevan, in the “ Philosophical Transactions ” of 1829, p. 127. As these may interest the members of the Aeronautical Society I give below a list of his results. I have, however, taken the liberty of substituting my own records for the average weights per cubic foot, as Bevan gives but few, and these to my mind are unsatisfactory, being the result of single trials, whereas mine are the mean of series of tests collected from all sources. I have then rearranged the list according to these weights in order to test Bevan's suggestion, that “the modulus of torsion bears a near relation to the weight of the wood.” Certainly the higher results are found amongst the heavier woods; there are, however, a few exceptions that render a blind confidence in this rule somewhat dangerous and it should be accepted as a rough guide only. I regard it more as a demonstration of the very general fact that, apart from freaks of structure, heavy woods are in all respects stronger than light woods.