264 



VII. DYNAMICS OF ROTATING BODIES. 



been proved to have no points of inflexion, and is like Fig. 78, which 

 has been calculated for A = 8, B = 5, C = 3, ^ = 4 - 9. 



86. Stability of Axes. We have seen that the body if 

 rotating about either of the principal axes of inertia will remain 

 rotating about it. If the instantaneous axis be the axis of either 

 greatest or least inertia, and be displaced a little, as the polhodes 

 encircle the ends of these axes the instantaneous axis will travel 



around on a small polhode, 

 and the herpolhode will be 

 small, neither ever leaving the 

 original axis by a large amount. 

 These axes are accordingly 

 said to be axes of stable motion. 

 If on the other hand the mean 

 axis be the instantaneous axis, 

 and there is a slight displace- 

 ment, the axis immediately 

 begins to go farther and farther 

 from the original position, and 

 nearly reaches a point diame- 

 trically opposite before return- 

 ing to the original position. 



pig 78< The mean axis is thus said to 



be an axis of instability. It is 



however to be noticed that if either A B or B C is small with 

 respect to the other, the separating polhode closes up about either 

 the axis of greatest or least inertia respectively, and thus a small 

 displacement may lead to a considerable departure from the original 

 pole, the rotation is thus less stable. The rotation about either axis 

 is most stable when the wedge of the separating polhode enclosing 

 it is most open. 



87. Projections of the Polhode. From the equations of the 

 polhode 37), 39), we may obtain its projections on the coordinate 

 planes by eliminating either of the coordinates. Eliminating x, 



45) 



an ellipse of semi -axes, 



B(A-B) S 



the ratio of which is 



yc(A-c\ 



