Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/342

position of the propeller, the essential point being the extent to which the frictional wake is led into the periptery of the propeller blades. It is manifest that ${\mbox{v}}_{1}$ is limited to a value less than the sternward component of the impressed velocity (compare § 200).

In the hypothetical case chosen in § 200, where it is assumed that the whole of the wake current is utilised by the propeller, we have—

Phantom ship velocity $={\mbox{V – v}}_{1},$ or by § 198—

$E={\frac {{\mbox{V – v}}_{1}}{{\mbox{V – v}}_{1}+{\frac {\mbox{v}}{2}}}}=$ ${\frac {{\mbox{V – v}}_{1}}{{\mbox{V –}}\ {\frac {{\mbox{v}}_{1}}{2}}}},\quad ({\mbox{for}}\ {\mbox{v}}_{1}={\mbox{v}}),$

but for real ship

$E_{1}={\frac {\mbox{V}}{{\mbox{V – v}}_{1}}}\ E,$

or

$E_{1}={\frac {\mbox{V}}{{\mbox{V – v}}_{1}}}\times {\frac {{\mbox{V – v}}_{1}}{{\mbox{V –}}\ {\frac {{\mbox{v}}_{1}}{2}}}}={\frac {\mbox{V}}{{\mbox{V –}}\ {\frac {{\mbox{v}}_{1}}{2}}}},$

which is the result already deduced in § 200 by the direct application of the Newtonian principle.

The device of the phantom ship is in reality merely a method of expressing a simple problem in relative motion in a palatable form; it is obvious that the argument treats the wake current as a favourable tidal current, or as the flow of a river, the ship's motion being credited in respect of its change of position relatively to some fixed mark; the method of the “phantom ship” presents the problem in a clear and precise form.

The question of wake influence is probably of less importance in connection with aerial flight than it is in the problem of marine propulsion.

§ 217. The Hydrodynamic Standpoint. Superposed Cyclic Systems.—It is of interest to form a mental picture of the hydrodynamic system of flow that accompanies a screw propeller.

It is evident that according to peripteral theory each blade of the propeller forms the axial core of a cyclic system and that the

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