Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/289

and further, when flexed by the pressure to which they are subjected in flight it is probable that they actually present a convex surface to the “wind.” On the contrary, the wings of birds are always concave on the under side and convex above; they are in fact true pterygoid forms. This is not only the case when the wing is quiescent but is visibly the case when the bird is in flight. It is of particular interest that some of the larger butterflies and moths—for example, many of the ornithoptera—show clearly a rudimentary development of the dipping front edge, proving that this feature is not merely an incident of a different method of construction.

§ 185. The Weight per Unit Area as related to the Best Value of $\beta .$—We may now resume the main subject from the point to which it was carried in § 181, and we can show that the value of $\beta$ corresponding to a minimum gliding angle denotes a definite relationship between the area $A$, the velocity $V,$ and the load carried $W.$

According to the hypothesis of constant sweep, we know that the mass dealt with per second is given by the expression $\rho \ \kappa \ A\ V,$^[1] and the velocity of the up-current is a $V,$ and that of downward discharge $=\beta \ V,$ on the assumption that we are dealing with small angles.

Consequently the weight supported ($W$) which is equal to the momentum communicated per second, will be $\rho \ \kappa \ A\ V^{2}\ (\alpha +\beta );$ but we have $\alpha =\epsilon \ \beta ,$ so that our expression becomes—

$W=\rho \ \kappa \ A\ V^{2}\ \beta \ (\epsilon +1),$

or—

${\frac {W}{A\ V^{2}}}=\rho \ \kappa \ \beta \ (\epsilon +1),$ which is constant.

Now ${\frac {W}{A\ V^{2}}}$ may be written ${\frac {P_{2}}{V^{2}}}$ where $P_{2}$ denotes pressure, i.e., weight per unit area sustained by the aerofoil. In Table IX. are given values of ${\frac {P_{2}}{V^{2}}}$ for aerofoil of pterygoid form and of different aspect ratio, calculated from values of $\beta$ given in

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1. An erratum published in Volume 2 has been applied: "P. 269, line 12 from foot, for 'k' read '$\kappa .$'" (Wikisource contributor note)