Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/319

Efficiency $=1-{\frac {1}{\cot \gamma \ \cos \theta \ \sin \theta -\sin ^{2}\theta +1}}$

From which by transformation,

Efficiency $={\frac {\tan \theta }{\tan(\theta +\gamma )}}.$

This expression may be deduced directly from the conditions. Let Fig. 125 represent, by the lines $O\ a$ and $O\ b,$ the helices of horizontal and gliding path respectively; then, since the reaction ${\mbox{W}}$ is normal to the path $O\ a,$ the work done when the propeller is rotated through an angle represented by the line $O\ d$ will be ${\mbox{F}}\times b\ d;$
Fig. 125. but the work utilised is represented by ${\mbox{F}}\times b\ d;$ the efficiency is therefore ${\frac {b\ d}{a\ d}},$ that is ${\frac {\tan \theta }{\tan(\theta +\gamma )}}.$

The result is thus obtained in a more direct manner, all trigonometrical transformations being dispensed with; the original demonstration is, however, of a more exact nature and is based on a clearer conception of the conditions involved. The identity of the two methods may be demonstrated geometrically by showing that the shaded areas of Fig. 124 are proportional to the ordinates $b\ d$ and $b\ a$ of Fig. 125, a matter of perfect simplicity.

The present theory enables us to define the slip of the propeller as the difference between the ordinates $b\ d$ and $a\ d,$ the slip ratio being represented by ${\frac {a\ b}{a\ d}}.$ The term slip as here defined is not identical with the slip of the naval architect, which is derived from the mean pitch angle of the blades, a basis that can have no justification in theory. The conception of slip originated with the propeller of true helical form and then denoted the difference between the geometrical and effective pitch; when blades were given an increasing pitch the mean pitch angle was taken as the basis of calculation of the geometrical pitch; hence the present

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