Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/261

In the above illustration we have assumed that the bird has been correctly designed by Nature on the basis of oar present hypothesis. There is an item of some importance which we have hitherto neglected and which will be subsequently taken into account, i.e., the influence of sail area on total weight. We have so far assumed that the weight is constant and that the sail area may be increased or diminished at will, whereas in reality a part of the total weight is due to the wings themselves, and the total weight should be represented as some function of the area (F) A, plus a constant. This extension of the subject will be left for later investigation; for the present we will continue to exhaust the problem from the present standpoint.

§ 168. Relation of Velocity of Design to Velocity of Least Energy. It has been pointed out in respect of props, i. and iv., in Cor. II., prop, v., that the velocity for which an aerodrome is correctly designed to cover the greatest distance on a given supply of energy is not the velocity at which it will actually cover the greatest distance, unless the body resistance is zero. Let us put the matter in the form of a further proposition:—

Prop. VI.—Given the relation of $x_{1}$ to $x_{2},$ determine the velocity of least resistance in terms of the velocity for which the aerodrome is designed for least resistance.

Let us represent the designed velocity by the symbol ${\mbox{V}},$ and let $V_{1}$ (as in prop, iii.) represent the velocity of least resistance, that is, when $x=y$ (prop. i.). Then at velocity ${\mbox{V}}$ we have $x_{2}=y,$ and at velocity $V_{1}$ we have $x_{1}+x_{2}=y,$ where $x$ and $y$ are variables.

Let $a_{1}$ and $a_{2}$ represent normal areas that will give rise to resistances equal to $x_{1}$ and $x_{2}.$

Then,	$x_{1}=n\ a_{1}\ V^{2}$
	$x_{2}=n\ a_{2}\ V^{2}$ where $n$ is a const.
and,	$y=m\ {\frac {1}{V^{2}}}$ where $m$ is a const.
When	$V={\mbox{V}},\quad x_{2}=y,$