Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/228

§ 150 obvious that in apteroid aspect the augmentation will be very small indeed, and if we go so far as to suppose the plane of infinite length, then the augmentation vanishes.

Thus the infinite parallel lamina, in apteroid aspect, affords the case of a plane that will conform to the sine2 law, and the pressure on its faces will be given by the expression: $$P_\beta = P_{90} \times \sin^2 \beta ,$$ or in full: $$P_\beta = C\ \rho\ V^2\ \sin^2 \beta$$ where $$C$$ is the constant of the normal plane, or in absolute units for a plane of the form under discussion in air, $$C = .76\ \mbox{or}\ .78$$ (about).

The above result becomes self-evident from the point of view of relative motion. The conditions of the problem will be fully represented if we suppose an infinite parallel lamina in normal presentation to slide along in the direction of its own length. It is evident that such sliding motion, presuming no skin-friction, can have no effect whatever upon the pressure reaction, and therefore by § 145, the sine^ law holds good.

We might go so far as to suppose the above experiment to be tried on a “whirling table” (Chap. X.), the plane being extended to form a complete ring bounded by two concentric circles. Assuming the method to be that of the falling plane, it is evident that the time of fall of such a ring will be substantially independent of its velocity of rotation.

§ 151. Planes in Apteroid Aspect (Experimental).—In Fig. 95 we have plotted to a common maximum value: (A) the curve of sine2 as deduced in the preceding article for the special case where the plane extends to infinity; (B) the Duchemin curve for the square plane, the Dines curve also being shown dotted; (E) curve as plotted by Langley for plane, 6 inches by 24 inches; (F) curve as plotted by Dines for plane, 3 inches by 48 inches. If, as there is every reason to suppose, the normal pressure is a continuous function of the aspect ratio of the plane, then as we suppose the latter to undergo variation from the square to the infinite