Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/221

Rh whole form, is of necessity unreliable and in general entirely misleading.

An aeroplane may be regarded as the special case of a body whose whole form is defined by the shape of its face in presentation, and consequently in stated aspect its pressure reaction can be expressed as a function of its angle and velocity.

§ 145. The Sine2 Law of Newton.—The difference between the behaviour of a real fluid and the Newtonian medium, sufficiently evident in the case of the normal plane, is further accentuated when the effect of inclining the plane is taken into account.

According to the hypothesis of the Newtonian medium the pressure is due to the impact of the particles of which the medium is composed. In the present case it is simplest to presume, in the first instance, that the plane and particles are perfectly elastic. Let Fig. 92 represent a plane the pressure on which is due to the momentum communicated by a Newtonian medium, whose relative path is that indicated by the arrows making an angle $$\beta$$ with the plane itself. Then if $$A$$ be the area of the plane, and $$V$$ the velocity of the “medium” whose density is $$\rho ,$$ the total momentum of the stream per second $$= \rho\ A\ V^2 \sin \beta,$$ and component normal to plane, $$= \rho\ A\ V^2 \sin^2 \beta ,$$ and on the assumption of perfect elasticity the total momentum communicated per second is:—$$2\ \rho\ A\ V^2 \sin^2 \beta ,$$ or, if $$P_\beta =$$ pressure per unit area on the plane we have—

Rh

But for the normal plane, denoting the pressure by the