Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/260

by a component of gravity (Fig. 110), and if $\gamma$ be the angle of descent, we have:—

${\frac {\mathit {Thrust}}{\mathit {Weight}}}=\sin \gamma ,$

that is to say, $\gamma$ is constant. If we take account of the body resistance $x_{1}$, we shall find that the value of $\gamma$ will increase the higher the velocity. This effect is more fully investigated in the subsequent section.
Fig. 110.

§ 167. The Gliding Angle, as affected by Body Resistance.—Let $\gamma ,$ as before, stand for the theoretical (constant) gliding angle when the body resistance is zero.

Let $\gamma _{1}$ be the gliding angle when the body resistance is $x_{1};$ then:—

When the total resistance is $x_{2}+y$ gliding angle $=\gamma ,$ and when total resistance is $x_{1}+x_{2}+y$ gliding angle $=\gamma _{1},$ the weight $W$ being the same in both cases.

Consequently we have for small angles ${\frac {\gamma _{1}}{\gamma }}={\frac {x_{1}+x_{2}+y}{x_{2}+y}},$ which for a correctly designed aerodrome, when $x_{2}=y,$ becomes—

${\frac {\gamma _{1}}{\gamma }}={\frac {x_{1}+2\ x_{2}}{2\ x_{2}}},$

Taking as an illustration the case of a bird, and estimating the relation of $x_{1}$ to $x_{2}$ on the basis of skin-friction alone (which is probably near the truth), we find by measurement of different species that the body surface is at least $\scriptstyle {\frac {1}{3}}$ of the wing surface, that is to say, we may take it that $x_{2}=3\ x_{1}$ and we have—

${\frac {\gamma _{1}}{\gamma }}={\frac {7}{6}};$

that is to say, under the most favourable circumstances in bird flight the gliding angle is increased about $\scriptstyle {\frac {1}{6}}$ by body resistance above what it would be were such resistance absent. In most game birds and other fast fliers the proportion would work out very much higher.

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