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

(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 $x\propto {\frac {1}{y}}$

∴

${\frac {dx}{dy}}={\boldsymbol {-}}{\frac {x}{y}}$

(1)

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

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 $Y=3\ X$

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:—

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