Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/255

Rh (1) To determine the conditions under which the greatest distance may be covered on a given supply of energy that is to say, the conditions of least resistance;

(2) To remain in the air for the longest possible time on a given supply of energy, that is, to determine the conditions of least horse-power.

Prop. I.—We have—

By § 157, $$x \propto V^2$$ and by § 159 (5), $$y \propto \frac{1}{V^2}$$ or $$x \propto \frac{1}{y}$$

Now conditions are fulfilled when $$x + y$$ is minimum, that is when $$dx = \boldsymbol{-} dy ,$$ or $$\frac{dx}{dy} = \boldsymbol{-} 1 ,$$   by (1), $$\ \boldsymbol{-} \frac{x}{y} = \boldsymbol{-} 1 ,$$

or,

Therefore, under the conditions of hypothesis, an aerodrome will travel the greatest distance on a given supply of energy when its aerodynamic and direct resistances are equal to one another.

Prop. II.—We have—

$$xV$$ power expended (energy per second) in overcoming direct resistance.

$$yV$$ power expended (energy per second) in overcoming aerodynamic resistance.

Then $$xV \propto V^3$$ and $$yV \propto \frac{1}{V}$$

Denote $$xV$$ by $$X$$ and $$yV$$ by $$Y,$$ we have $$X \propto \frac{1}{Y^3}$$ or, $$\frac{dX}{dY} = \boldsymbol{-} 3 \frac{X}{Y} ,$$ and when $$dX = \boldsymbol{-} dY$$ we have $$Y = 3\ X$$

or,

Therefore, an aerodrome will remain in the air for the longest possible time on a given supply of energy, that is to say, its fight will be accomplished on least horse-power, when the resistance due to aerodynamic support is three times the direct resistance.

On the foregoing propositions a third may be founded as follows:—