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

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

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.

§ 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—

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 of the wing surface, that is to say, we may take it that $$x_2 = 3\ x_1$$ and we have—

that is to say, under the most favourable circumstances in bird flight the gliding angle is increased about 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.