and with a thickness in the centre of 7/16 inch, I obtained results almost identical with those of the very much thinner brass aeroplane, but it must not be supposed that in practice an aeroplane is completely without friction. If it is very rough, irregular in shape, and has any projections whatsoever on either the top or bottom side, there will be a good deal of friction, although it may not, strictly speaking, be skin friction; still, it will absorb the power, and the coefficient of this friction may be anything from ·05 to ·40. These experiments with the brass aeroplane demonstrated that the lifting effect was in direct proportion to the angle, and that skin friction, if it exists at all, was extremely small, but this does not agree with a certain kind of reasoning which can be made very plausible and is consequently generally accepted. [Illustration: Fig. 2.--Professor Langley’s experiments--_a_, end of the rotating arm; _b_, brass plane weighing 1 lb.; _c_ _c_, spiral springs. When the arm was driven through the air, in the direction shown, the plane assumed approximately a horizontal position, and the pull on the springs _c_ _c_ was reduced from 1 lb. to 1 oz.] Writers of books, as a rule, have always supposed that the lifting effect of an aeroplane was not in proportion to its inclination, but in proportion to the square of the sine of the angle. In order to make this matter clear, I will explain. Suppose that an aeroplane is 20 inches wide and the front edge is raised 1 inch above the horizontal. In ordinary parlance this is, of course, called an inclination of 1 in 20, but mathematicians approach it from a different standpoint. They regard the width of the aeroplane as unity or the radius, and the 1 inch that the front edge is raised as a fraction of unity. The geometrical name of this 1 inch is the sine of the angle--that is, it is the sine of the angle at which the aeroplane is raised above the horizontal. Suppose, now, that we have another identical aeroplane and we raise the front edge 2 inches above the horizontal. It is very evident that, under these conditions, the sine of the angle will be twice as much, and that the square of the sine of the angle will be four times as great. All the early mathematicians, and some of those of the present day, imagine that the lift must be in proportion to the square of the sine of the angle. They reason it out as follows:--If an aeroplane is forced through the air at a given velocity, the aeroplane in which the sine of the angle is 2 inches will push the air down with twice as great a velocity as the one in which the sine of the angle is only 1 inch, and as the force of the wind blowing against a normal plane increases as the square of the velocity, the same law holds good in driving a normal plane through still air. From this reasoning, one is led to suppose that an aeroplane set at an angle of 1 in 10 will lift four times as much as one in which