66 Mr. J. W. Sharpe on the Boomerang. 



motion from convexity of the surfaces; but otherwise the 

 analysis given above for the case of the boomerang will 

 suffice also for this case. In both cases the m omental ellip- 

 soid at the C.G. has its least axis normal to the principal 

 plane ; and owing to the thinness of the boomerang this axis 

 is very much smaller than the other two. The boomerang, 

 but not the model, is thrown w r ith the axis of angular 

 momentum, and therefore also the instantaneous axis of rota- 

 tion, initially coincident with this least axis of the momental 

 ellipsoid : and, in the absence of the air, these three axes would 

 remain coincident, and their orientation would remain un- 

 changed, throughout the. motion. The actual motion, however, 

 as disturbed by the reaction of the air, i. e. the rotational 

 motion, is very stable, by reason of the relative smallness 

 of the minimum axis of the momental ellipsoid at the C.G., 

 and the high rate of spin imparted by the thrower. Those 

 disturbing air couples which are in planes parallel to the 

 principal plane merely reduce the angular momentum with- 

 out deflecting its axis; but all the other couples, in planes, 

 that is, parallel to the axis of the boomerang, effect in each 

 revolution a very slight displacement of this axis without 

 appreciably altering the actual amount of the angular mo- 

 mentum, or that of the kinetic energy of the rotational 

 motion, because their axes are at right angles to the axis of 

 angular momentum, or, at all events, very nearly so. The 

 invariable plane always touches the momental ellipsoid very 

 near the extremity of the least axis*, and the C.G. remains 

 very nearly at a constant distance from this plane, and the 

 ellipsoid is there much flattened. Consequently the instan- 

 taneous axis, the axis of the boomerang, and the axis of 

 angular momentum, are always nearly coincident, the two 

 former following the displacement of the latter. 



In conclusion it is interesting to note that similar actions 

 on the part of the air cause rifle-bullets and conical shells to 

 set themselves tangentialiy to the path of their C.G., instead 

 of keeping their axes unchanged in orientation throughout 

 their motion, which would be the ordinary effect of the co- 

 incidence of the axis of figure with that of angular momentum. 

 Take the case of a bullet fired from a barrel with right- 

 handed rifling. The result is just the same with left-handed 

 rifling. The C.G. and the axis of angular momentum are in 

 the axis of figure, the latter being directed forward. Now, 

 owing to the downward concavity of the path of the C.G., 

 the bullet tends continually to meet the air more and more 



* Because the angular momentum and the kinetic energy of the 

 rotational motion remain nearly constant. 



