Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/256

Prop. III.—To determine the relation of the speed of greatest range to the speed of least power.

Now $x=kV^{2}$ and $y=n\ {\frac {1}{V^{2}}},$ where $k$ and $n$ are constants.

When $x=y$  let $\ V=V_{1}.$

When $3\ x=y$  let $\ V=V_{2}.$ It is required to find the relation of $V_{1}$ to $V_{2}.$

When $x=y$  we have $k\ V_{1}^{2}={\frac {n}{V_{1}^{2}}}$  or $\ k={\frac {n}{V_{1}^{4}}}.$

When $3\ x=y$  we have $\ 3\ k\ V_{2}^{2}={\frac {n}{V_{2}^{2}}}.$

Substituting for $k$ we have—

$3\ {\frac {n}{V_{1}^{4}}}\ V_{2}^{2}={\frac {n}{V_{2}^{2}}},$  or $\ 3={\frac {V_{1}^{4}}{V_{2}^{4}}}$  or $\ {\frac {V_{1}}{V_{2}}}=3^{\frac {1}{4}}=1.315$

That is to say, the speed of greatest range is 1.315 times the speed of least power.

Corollary to Prop. III.—For a plane aerofoil the change in value of the angle $\beta$ involved in the change of velocity from $V_{1}$ to $V_{2}$ can be immediately deduced.

$V_{1}=3^{\frac {1}{4}}\ V_{2},$ but by § 159, $V^{2}\ \beta \propto W$ (for small angles). Consequently $V_{1}^{2}\ \beta _{1}=V_{2}^{2}\ \beta _{2}$ where $\beta _{1}$ and $\beta _{2}$ are the angles appropriate to the velocities $V_{1}$ and $V_{2}$ respectively. Therefore—

$\beta _{1}\ (3^{\frac {1}{4}})^{2}\ V_{2}^{2}=\beta _{2}\ V_{2}^{2}$

Thus calculations of $\beta$ values for least resistance require to be multiplied by ${\sqrt {3}}$ to give appropriate values for least horsepower. We may thus anticipate that birds whose object in flight is to fall as slowly as possible (as birds whose habit is to be sustained on an upcurrent, and so to take advantage of the least upward velocity possible), will have wings of hollower form than those whose object is to get from point to point.

§ 165. Examination of Hypothesis.—According to the hypothesis on which the foregoing propositions are founded, it is supposed

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