RIGID BODY ABOUT A FIXED POINT. 299 



ib = — ax — by cos nt — bz sin nt , 

 x = aw + bz cos nt — by sin nt , 

 y = bw cos nt + bx sin w£ — az , 

 z = bw sin nt + ay — bx cos w£ . 



By substitution in these the above result may be verified. 



53. Consider, as an example of applied forces, a homogeneous solid of revolu- 

 tion moving about a fixed point in its axis, which is not its centre of gravity. 

 To determine the motion. 



If a, a unit- vector, represent at time t the position of the axis of the solid, we 

 may choose the tensor of 7, a vertical vector, so that the couple due to gravity is 

 Yay. Hence the equation of motion is §§ 24, 22, 



<pi + Yspe = Yay . 



But 



<p s = Bg — (A — B) a Sag , 



so that 



Be - (A - B) aSae - (A - B) V«aSa« = Yay . . . (44). 



This, with the kinematical relation 



a = Vsa ...... (1), 



contains the complete solution of the problem. 

 54. Operating on (44) by S . a, we have 



Sal = . 



But, by (1), we have 



Sas = . 



Hence 



Sas = constant = Q (45) 



(that is, the angular velocity about the axis of revolution of the solid is constant) 

 and (44) is reduced to the form 



Bi - (A - B) Qa = Yay (46). 



But, by (45) and (1), 



ea = Q + a , 



or 



s = — Qa + ad. . . . . . . (47). 



Since aa is a vector, we have (as in § 30) 



(48), 



