Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/337

Referring to § 181, we have the angle $\beta$ given by the expression ${\sqrt {\frac {2\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}}.$ Taking $\xi$ for water as = .01 and density = 64 lbs. per cubic foot we obtain, $\beta$ = .132 (radians) or, = 7.6°, and the theoretical minimum gliding angle is 3.95°, which in practice becomes 6° about. The $P_{2}/V^{2}$ relation (given for air in Table IX.) will become $\rho \ \kappa \ (\epsilon +1)\ \beta =$ 12.5.

The above are the data on the pterygoid basis; similarly on the plane basis, that is, for blades of perfect helical form, we have, taking $\xi =$ .005, on the principle explained in § 182, $\beta =$ .048, or, $\beta$° 2.75°, that is $\gamma$ (least value) = .096, or $\gamma$° = 5.5°. The $P_{3}/V^{2}$ relation is given by the expression—
${\frac {P_{3}}{V^{2}}}=c\ \beta \ C\ \rho$ (§ 186) = 4.55.

The above results may be tabulated as follows:—

Pterygoid basis.	${\begin{cases}\\\\\\\\\\\\\end{cases}}$	$\beta ^{\circ }$	.  .  .  .  .  .	7.6°
		$\gamma ^{\circ }\ {\bigg \{}$	(calculated)	3.95°
			(probable)	6°
		${\frac {P_{2}}{V^{2}}}$	.  .  .  .  .  .	12.5°
Plane basis.	${\begin{cases}\\\\\\\\\\\end{cases}}$	$\beta ^{\circ }$	.  .  .  .  .  .	2.75°
		$\gamma ^{\circ }$	.  .  .  .  .  .	5.52°
		${\frac {P_{3}}{V^{2}}}$	.  .  .  .  .  .	4.55°

The $P/V^{2}$ values given above are in absolute units. The pressure value is extended in Table XIV. as pounds per square foot for values of $V$ in feet per second.

§ 215. The Marine Propeller (continued).—Cavitation.—It is very questionable to what extent the plane basis of operation is applicable in the case of a screw propeller. There seems to be a grave theoretical objection that does not exist when the motion is rectilinear, as in the analogous case of a weight supported by an aeroplane.

When we have to deal with the propeller blade on the

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