Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/258

Thus, if for the given conditions of weight and velocity the aerofoil be made too large, the skin friction will be in excess of the aerodynamic resistance; if insufficient surface is provided, the aerodynamic resistance will be in excess; in neither case will the energy required for a given distance be the least possible.

It may be noticed that this result is in appearance out of harmony with prop, i., for there it was shown that the most favourable velocity at which to run an aerodrome in order to cover the greatest distance on a given quantity of energy is that at which the aerodynamic resistance is equal to the total direct resistance, that is $x;$ whereas according to the present proposition the most economical conditions are met with when $y=x_{2},$ which is only a portion of the total.

The explanation of this apparent paradox will be given in the light of the subsequent proposition.

§ 166. Velocity and Area both Variable.—Prop. V.—Given that $x_{1}=0,$ then, for an aerodrome of given weight, with $V$ and $A$ both variable, find the velocity at which a given flight (distance) can be accomplished with least energy.

By § 159 (5a)	$y\propto {\frac {1}{AV^{2}}}$
and,	$x_{2}\propto AV^{2}$
Now for $x_{2}+y$ minimum, we have $x_{2}=y$ (by Prop. IV.), or,—
	$AV^{2}\propto {\frac {1}{AV^{2}}}$
∴	$(AV^{2})^{2}=$ constant.
hence,	$y=$ constant.
and,	$x_{2}=$ constant.
∴	$x_{2}+y$ is constant,

or the resistance is independent of the velocity.

That is to say, if for an aerodrome of given weight the velocity be supposed to undergo continuous variation, and the “sail area” also undergo corresponding variation, so that the latter is at every

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