forward velocity--that is, in moving 10 feet, it pushes the air down 1 foot. A good many mathematicians rely altogether upon the acceleration of the mass of air beneath the aeroplane which is accelerated by its march through the air, the value of this acceleration being in proportion to the square of the velocity which is imparted to it. Suppose now that the aeroplane is thin and well-made, that both top and bottom sides are equally smooth and perfect; not only does the air engaged by the under side shoot downwards, but the air also follows the exact contour of the top side, and is also shot downwards with the same mean velocity as that passing on the underneath side, so if we are going to consider the lifting effect of the aeroplane, we must not leave out of the equation, the air above the aeroplane, which has quite as much mass and the same acceleration imparted to it, as the air below the aeroplane. Even calculations made on this basis will not bring the lifting effect of an aeroplane up to what it actually does lift in practice; in fact, the few mathematicians who have made experiments themselves have referred to the actual lifting effect of aeroplanes placed at a low angle and travelling at a high velocity as being unaccountable. Only a few mathematicians appear to have a proper grasp of the subject. However, three could be pointed out who understand the subject thoroughly, but these are all mathematicians of the very highest order--Lord Kelvin, Lord Rayleigh, and Professor Langley. In placing before the public, the results of my experiments and the conclusions arrived at, it is necessary to show the apparatus which I employed, otherwise it might be inferred that my conclusions were guesswork, or mathematical calculations which might or might not be founded on a mistaken hypothesis; this is my excuse for showing my boiler and engine, my rotating arm, and my large machine. I do not anticipate that anyone will ever use a steam engine again, because any form of a boiler is heavy; moreover, the amount of fuel required is much greater than with an internal combustion engine, and certainly seven times as much water has to be dealt with. However, the description which I am giving of my apparatus will demonstrate that I had the instruments for doing the experimental work that I have described in this work. In the Appendix will be found a description of my machine, and some of my apparatus. The conclusions which I arrived at were written down at the time with a considerable degree of care, and are of interest because they show that, at that date, I had produced a machine that lifted considerably more than its own weight and had all of the essential elements, as far as superposed aeroplanes, fore and aft horizontal rudders, and screw propellers were concerned, common to all of the successful machines which have since been made. The fact that practically no essential departure has been made from my original lines, indicates to my mind