Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/273

The resistances, and therefore the gliding angles, may be presented in the form of a diagram (Fig. 111), in which abscissae represent velocity and ordinates the gliding angle; the dotted line represents the constant resistance, and the curve (struck from the dotted line as datum to the equation $\gamma \propto V^{2}$) shows the manner in which the resistance increases with the velocity. Values of $V$ and $\gamma$ have been assigned for a supposititious case.

§ 176. Value of $\beta$ and $\gamma$ for Least Horse-power.—By prop, ii., § 164, we know that the condition for least horse-power is—$y=3x,$ when $y=3x$ let $\beta =\beta _{2}.$

Then, following § 173—

${\frac {1}{2}}\ \kappa \ \beta _{2}^{2}\ (1-\epsilon ^{2})=3\ \xi \ C$

or

$\beta _{2}^{2}={\frac {6\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}$

$\beta _{2}={\sqrt {\frac {6\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}}.$

A result that otherwise follows from corollary to prop. iii.—

$\beta _{2}={\sqrt {3\ \beta _{1}}}.$

Let $\gamma _{2}=$ gliding angle for least horse-power. Following § 174 we have—

	$\gamma _{2}={\frac {y+x}{W}}$  where  $y=3\ x$
or,	$\gamma _{2}=1{\frac {1}{3}}\ {\frac {W}{y}}=1{\frac {1}{3}}\ {\frac {\beta _{2}^{2}-\alpha ^{2}}{2\ (\beta _{2}+\alpha )}}$
∴	$\gamma _{2}={\frac {2}{3}}\ \beta _{2}\ (1-\epsilon )$
or in terms of $\beta _{1}$—
	$\gamma _{2}={\frac {2}{\sqrt {3}}}\ \beta _{1}\ (1-\epsilon )$
or,	$\gamma _{2}={\frac {2}{\sqrt {3}}}\ \gamma _{1}=1.155\ \gamma _{1}.$ (approx.).

In Fig. 112 the $x$ and $y$ resistances are shown as curves separately and superposed. In the lower portion of the figure

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