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

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^[1] 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 $\beta$—.

Let	$\epsilon =$	${\frac {\alpha }{\beta }}$ and, as before,
„	$A=$	aerofoil area,
„	$\kappa A=$	sweep,
„	$\xi =$	coefficient of skin friction.

$C$ 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

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

1. The values of $c$ for the aeroplane are probably not the same as for a pterygoid aerofoil of the same aspect ratio. Neither value has yet been determined with any degree of accuracy; the values given in Figs. 105 and 106 and in tabular form in § 177 are probably more nearly correct for the pterygoid form.
   It will be shown subsequently that the effective value of $\xi$ in the case of an inclined plane may be less than its true value.

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