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.

The present decade will pass down into history notorious for many things good and bad. It will be styled a time of improvement in many ways. It will be remembered as covering the period of the world's greatest war, but it will also be remembered among all trades appertaining in any way to engineering as the age of progress in the manufacture and manipulation of steel tubing.

Independent altogether of the stimulus given to the use of tubing for structural work by the advent and advance of aeronautics, the uses to which tubing of steel and other metals is put has increased considerably during the past few years. Much, of course, has been due to the large output of cycles and motor-cycles, but in addition an immense quantity of tubing is used in the construction of motorcars, and in a thousand other productions which it is unnecessary to particularise in this paper.

It has long been realised that experiments in the laboratory on small models afford in certain cases a means of obtaining, cheaply and rapidly, information on the behaviour of the actual structure under the conditions in which it is to be used. This plan has been adopted in many branches of science, and in none to a larger extent than in naval and aeronautical architecture. The problems to be investigated are usually those connected with the motion of bodies through a fluid, but if we accept the principle of relative motion we have an alternative method open to us for our model experiments. We may, if we wish, reverse the conditions and set the fluid in motion, the model remaining stationary. Generally speaking, one plan has been adopted in naval architecture and the other in aeronautics. In the former case models are towed through water in a ship tank, while in the latter they are placed in a tube through which air is passed.

The question of “relativity” is much to the front at the present day, and the methods of viewing it may be divided, very roughly, into three classes:—

1.The relativity of space, time, motion, etc. Nothing being absolute— everything relative to something else. This view is very old (it certainly dates back as far as the time of Aristotle), and nearly everybody believes it; although there are still a few people who would like to believe in absolute motion, space, or rest. Even truth, however, is really “relative” ; and this was, apparently, Pilate's view when he scornfully asked, “What is truth?”

Gusts, says Duchêne, are more frequent when the machine is flying into the wind than with the wind, “for the simple reason that the gusts have a certain speed and the aeroplane therefore meets more of them (in the same space of time).” Elsewhere Duchêne refers to the experience of a pedestrian on a tramway route: in a given time a pedestrian meets more cars than those that overtake him.

I can see no parallel here, however, to meeting or following gusts in an aeroplane. Whether the machine is flying with the wind, or in the opposite direction to the stream of air, it has got to maintain its relative speed to the air: increase or decrease means a lift or a dip. So, as regards gusts in either case, the number met or overtaken would be the same. And it would seem that a following lull should have much the same effect on a machine as a head gust, and vice–versâ.

Ten years ago the aeroplane was a curiosity—a means wherewith, by a tour de force, a man might lift himself from the ground and make a hazardous flight through quiet air—a means to attract curious crowds to fenced-in county fair grounds to witness the marvel of a body heavier than air actually rising from the ground and moving under some measure of control through the air, and returning safely to its starting point.

What the aeroplane has become in this short but poignant decade of the world's history, and what it stands for to-day, this audience knows too well to need specification in detail.

It is perhaps well within the limits of conservatism to say that no achievement of man's inventive and constructive genius has undergone more intensive, more rapid, or more potentially significant development than has that of flying with an apparatus heavier than the air in which it moves.

The history and development of mechanical flight may be confined almost entirely to the present century. In the last century some success had been achieved in gliding flight, model aeroplanes had been made and flown, and at least two man-carrying aeroplanes had actually left the groundj but it was not until the year 1903 that the Wright Bros., having ended a painstaking research in gliding flight, earned for themselves the honour of being the first men to fly.

Directly the success of the Wright Bros, became known, or rather when people began to believe that what the Wright Bros, claimed was actually true, the development of flying became rapid in the extreme. To-day one is tempted to wonder why it was that it had taken man so many centuries of civilisation to do what now seems to most of us rather an ordinary thing.

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.

From Captain Hucks’ remarks on the difficulties of keeping a straight course when flying in clouds, and the impossibility of relying upon the compass as a guide when out of sight of a fixed point of reference, it would appear that the nature of the errors to which a compass is liable, and the circumstances under which it may be relied upon are not generally known.

Much work has been done at the Royal Aircraft Factory in the last three years on the behaviour of compasses in the air, and the directions to be followed in their design in order to avoid their errors as far as possible. This work was chiefly done by the late Captain Keith Lucas, F.R.S., who was unfortunately killed when learning to fly, in the autumn of last year. The following is a short statement as to what may be expected from compasses in aeroplanes:—

On account of the dip the N. point of the compass card, when an aeroplane is turning, always tends to point too much towards the centre of the turn, the amount of error varying according to the bearing from which the turn is made.

In this paper I propose to bring before you a broad outline of our knowledge of the subject, not from the point of view of a botanist, but purely from the point of view of an engineer.

Although in ancient, and in fact during all time, timber has been extensively used for constructional purposes, our knowledge of it is comparatively small, whilst the subject itself is very extensive.

The reason for these facts is not hard to find. Unfortunately the physicist has not in general had to deal with timber in the progress of physical research, as he has with metals; and the questions which concern the engineer are not in general those with which the botanist deals.

There has been comparatively little incentive for engineers to do much work on timber, as before the advent of the aeroplane its application was limited to unimportant parts, except in America and the Colonies, where the cost of metal parts is high. A considerable amount of information has in recent years been published by the American Government Forestry Department; the Government of New South Wales, Australia, have also published a considerable amount of data on Australian timbers.

In applying the methods of rigid dynamics of three dimensions to the study of aeroplane motions, one of the first questions which had to be settled by myself and Mr. Harper in the preparation of “Stability in Aviation” was the choice of co–ordinate axes and the notation to be adopted for the various forces, couples and rates of variation of components of air resistance. Where, as in this case, the necessary symbols more than exhaust the ordinary alphabet, the problem is a difficult one, and it was therefore intended to insert a separate section explaining the advantages which rendered the choice adopted the only logically possible one.

Unfortunately, at the time when “ Stability in Aviation “ was published, very great doubts existed as to the practical utility of any such investigation except as a subject of academic study in orthodox mathematical circles, and the work of preparing the book in the face of obstacles, partly arising from opposition, had proved such a strain that any further discussions on the subject would have been incompatible with my academic duties.