Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/269

Rh A plane of elongate form in pterygoid aspect whose value of $$c$$ is = 3 would thus have an angle of least resistance of slightly over 4$$\tfrac{3}{4}^\circ.$$ This is about the minimum value that would in the ordinary way be obtained, assuming that correct values have been assigned to $$c$$ and $$\xi .$$

When we have to deal with an aerofoil of curvilinear section adapted to the form of the lines of flow, we may obtain useful results by adopting the hypothesis of constant sweep (§ 160). According to this hypothesis it is assumed that the support is derived from a layer or stratum of fluid uniformly acted on by the aerofoil, and whose cross-sectional area is constant. This area, for a given plan-form of aerofoil in stated aspect, is equal to the aerofoil area $$A$$ multiplied by the constant $$\kappa$$, or, as given in § 160, we have, sweep $$= k\ A .$$

It will be further assumed that the relation $$\frac{\alpha}{\beta}$$ (§ 161) is constant for any given plan-form and aspect.

'''§ 173. The Pterygoid Aerofoil. Best Value of''' $$\beta$$—.

$$C$$ is the constant of the normal plane (§ 136).

Now the direct resistance $$x = \xi\ A\ C\ \rho\ V^2 ,$$ and the aerodynamic resistance $$y$$ is equal to the energy expended aerodynamically per second divided by the velocity, or

$$y = \frac{1}{2}\ \rho\ \kappa\ A\ V \times V^2\ (\beta^2\ \boldsymbol{-}\ \alpha^2)\ \mbox{÷}\ V$$ $$= \frac{1}{2}\ \rho\ \kappa\ A\ V^2\ (\beta^2\ \boldsymbol{-}\ \alpha^2) .$$