Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/257

that a constant weight is sustained at a varying velocity by an aerofoil of constant area, so that on the one hand the resistance due to skin-friction for any stated velocity undergoes no change, and on the other that the law $y\propto {\frac {1}{V^{2}}}$ shall be applicable, this law being that ascertained as pertaining to an aeroplane for small angles, and deduced generally in § 160 from the hypothesis of constant “sweep.”

So long as the foregoing hypothesis applies it is not important whether the direct resistance is entirely due to the skin friction of the aerofoil or whether it is in part due to the resistance of the “body” of the aerodrome, i.e., that part that may be supposed to constitute or contain the load. If we require to concern ourselves with changes of aerofoil or “sail” area, it becomes necessary to distinguish between these two kinds of resistance, the total resistance $x$ being supposed to be divided into two parts, the one $x_{1}$ being defined as independent of the sail area and the other $x_{2}$ as dependent and as directly proportional thereto. In all cases the approximate assumption is made that this class of resistance is proportional to velocity squared, the error that may result from this assumption being considered later.

Prop. IV.—To determine the conditions controlling the aerofoil area for an aerodrome of given weight travelling at a specified velocity.

Let $A=$ area, then, since $x_{1}$ is fixed by the conditions, $x_{2},y,$ and $A$ are the variables with which we are concerned:—

and as in prop. i. we have the minimum condition fulfilled when $x_{2}=y,$

Therefore the correct area has been given to the aerofoil when its aerodynamic resistance is equal to its direct resistance.

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