Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/275

Rh abscissae represent velocity and ordinates $$\gamma$$ values. It is evident that so long as we are confined to the small angle hypothesis the resistances may be thus represented and the sum of the separate $$\gamma$$ angles will give the resultant angle.

In the upper part of Fig. 112 the $$\gamma$$ angle is represented graphically, it being supposed that the aerodrome is launched from the point $$a$$.

Thus, if $$a\ b$$ represent the gliding angle of least resistance (shown for example as 10°) then $$a\ c$$ will represent the gliding angle for least power, the angle being 11°.55. If we suppose that two aerodromes are launched simultaneously from the point $$a$$ (of equal weight and “sail” area and plan-form), the one being designed for least resistance and the other for least power, their respective trajectories will be the two straight lines $$a\ b$$ and $$a\ c ,$$ and their positions after the lapse of a certain definite time will be given by the points $$b$$ and $$c$$ where $$a\ b$$ is to $$a\ c$$ in the relation $$\sqrt[4]{8} : 1$$ (prop, iii., § 164). We may draw a curve $$e\ c\ b\ d$$ through the points $$c$$ and $$b$$, which will represent the position occupied by aerodromes simultaneously launched from the point $$a$$, for other values of $$\beta .$$

Now since the angle of least resistance is the minimum gliding angle, the line $$a\ b$$ will be a tangent to the curve $$e\ b\ d$$ at the point $$b ,$$ and since the least power expenditure corresponds to the slowest rate of fall, the tangent to the curve at the point $$c$$ will be horizontal; we have thus defined the character of the curve in question, which represents the simultaneous loci for similar aerodromes of different $$\beta$$ values.

The existence or otherwise of body resistance does not affect the problem as here presented; it is included in the plotting as one of the resistances that vary as $$V^2 .$$

§ 177. The Values of the Constants.—The paucity of reliable data has already been made the subject of comment, and the values of many of the constants here given can only be regarded as rough approximations. To prevent misapprehension on this