Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/270

But condition of least total energy is that $x=y.$ Let $\beta =\beta _{1}$ and $\alpha =\alpha _{1}$ when $x=y.$

	$\xi \ C={\frac {1}{2}}\ \kappa \ (\beta _{1}^{2}-\alpha _{1}^{2})$	(1)
Now,	${\frac {\alpha }{\beta }}=\epsilon ,$  or  $\alpha _{1}^{2}=\beta _{1}^{2}\ \epsilon ^{2}$
or	$(\beta _{1}^{2}-\alpha _{1}^{2})=\beta _{1}^{2}\ (1-\epsilon ^{2})$
(1) becomes	$\beta _{1}^{2}\ (1-\epsilon ^{2})={\frac {2\ \xi \ C}{\kappa }}$
or	$\beta _{1}={\sqrt {\frac {2\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}}.$	(2)
If $\beta$ is expressed in degrees this becomes—
	$\beta _{1}^{\circ }={\frac {180}{\pi }}{\sqrt {\frac {2\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}}.$	(2a)

Of the quantities involved in this expression, $\xi$ and $C$ are known by experiment; $\kappa$ also may be experimentally determined by the method of superposed planes discussed in §§ 154 and 161; the experimental data are, however, at present wanting. For planes and other forms of aerofoil in pterygoid aspect $\kappa$ is a function of the aspect ratio and is greater when the aspect ratio is greater. The form of this function requires to be experimentally determined and plotted as a curve for certain simple geometrical plan-forms such as the ellipse and the rectangle, the co-ordinates to represent respectively the aspect ratio and the corresponding values of $\kappa .$

The quantity $\epsilon$ is also some function of the aspect ratio, and again we are lacking in experimental information. The values subsequently employed for $\kappa .$ and $\epsilon$ are those which the author is in the habit of using, and which are found to give results reasonably near the truth; they have not, however, been determined or verified by any scientific method, and must at present be regarded as open to suspicion.

It has been assumed in the present section that there is no loss of energy incidental to “handling” the fluid other than that due to skin friction. It is in practice possible that there is some unavoidable loss in eddy making by the aerofoil itself; especially

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