Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/222

symbol $P_{90},$ we know that for the conditions of the present hypothesis—

$P_{90}=2\ \rho \ V^{2}$ (§ 136),

or we have—

$P_{\beta }=P_{90}\times \sin ^{2}\beta .$

(2)

If we modify the hypothesis to the extent of supposing the plane inelastic both $P_{\beta }$ and $P_{90}$ are diminished in the ratio 2:1,
or,	$P_{\beta }=\rho \ V^{2}\sin ^{2}\beta ,$
and	$P_{90}=\rho \ V^{2},$
∴ the relation	$P_{\beta }=P_{90}\times \sin ^{2}\beta$	still holds good.

We may view this problem in another light with the same result. If we regard the motion of the plane as compounded of its edgewise and normal components, then the former can be neglected since it does not involve any reaction on the plane. Now if $V_{2}$ be the value of the normal component, the mass dealt with per second is $\rho \ A\ V_{1}$ and the momentum per second is $\rho \ A\ V_{1}^{2},$ or (on the elastic hypothesis),

$P_{\beta }=2\ \rho \ V_{1}^{2}$

(3)

which is the same as (1), for $V_{1}^{2}=V^{2}\ \sin ^{2}\beta .$

So that the pressure in the Newtonian medium is independent of the edgewise component of motion, and is the same as for a normal plane of velocity equal to the normal component of the actual motion.

An important consequence of this is that if we had to do with a Newtonian medium, or if a real fluid behaved as such, then the time of falling of a horizontal plane would be independent of any horizontal motion impressed upon it. The “falling plane,” therefore, becomes the experimentum crucis in respect of the “sine square” law.

§ 146. The Sine² Law not in Harmony with Experience.—It has long been known that in actual fluids the sine square law does not hold good. Probably the first experimenter to ascertain this fact was Vince in the year 1797 (Phil. Trans., 1798); later we find an explicit statement by Robinson (System of Mechanical

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