Page:Aerial Flight - Volume 2 - Aerodonetics - Frederick Lanchester - 1908.djvu/437

Hence the torque is diminished from this cause as well as that stated.

It appears to the author that the above facts point to an opening for the quantitative treatment of the problem. Thus, for example, if we confine ourselves to the portion of the curve. Fig. 200, where the displacement of the centre of pressure is sensibly constant,^[1] the torque that gives rise to the precession will, under changes of inclination, vary directly as the effective weight, that is inversely as cos a, where a is the angle made by the axis of rotation to the vertical (or by the plane of rotation to the horizontal). But by the equation to the gyroscope, App. VII,

$\tau =\mathrm {I} \omega \ \Omega \sin \alpha$

hence we can correlate changes in the angle a and the rate of precession. Thus in the particular case where a is 45° the torque and sin a have the same relative rate of change, and at this critical value the rate of precession Ω is constant in respect of a.

Fig. 203

It is well known that the flight of a boomerang is greatly assisted by an appropriate wind. In still air considerable skill is required to make sure of a clean return, but flights of the form given in Fig. 208 are quite easy even to a novice. When such flights are made with a wind blowing from the "left front" a defective throw will become a good return. If, for example, a given throw in still air would give rise to a path of roughly cycloidal form. Fig. 204 (a), we know that with an appropriate superposed translation this becomes an approximate circle. Fig. 204 (b), and likewise for

1. From an inspection of Fig. 200, and from the observed behaviour of a boomerang, it would appear that, for a considerable portion of the flight path at least, this assumption, as a first approximation, is justified.

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