Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/273

Rh The resistances, and therefore the gliding angles, may be presented in the form of a diagram (Fig. 111), in which abscissae represent velocity and ordinates the gliding angle; the dotted line represents the constant resistance, and the curve (struck from the dotted line as datum to the equation $$\gamma \propto V^2$$) shows the manner in which the resistance increases with the velocity. Values of $$V$$ and $$\gamma$$ have been assigned for a supposititious case.

§ 176. Value of $$\beta$$ and $$\gamma$$ for Least Horse-power.—By prop, ii., § 164, we know that the condition for least horse-power is—$$y = 3x ,$$ when $$y = 3x$$ let $$\beta = \beta_2 .$$

Then, following § 173—

$$\frac{1}{2}\ \kappa\ \beta_2^2\ (1 - \epsilon^2) = 3\ \xi\ C$$

or

$$\beta_2^2 = \frac{6\ \xi\ C}{\kappa\ (1 - \epsilon^2)}$$

$$\beta_2 = \sqrt{\frac{6\ \xi\ C}{\kappa\ (1 - \epsilon^2)} } .$$

A result that otherwise follows from corollary to prop. iii.—

$$\beta_2 = \sqrt{3\ \beta_1} .$$

Let $$\gamma_2 =$$ gliding angle for least horse-power. Following § 174 we have—

In Fig. 112 the $$x$$ and $$y$$ resistances are shown as curves separately and superposed. In the lower portion of the figure