274 VII. DYNAMICS OF ROTATING BODIES. 



is much less efficient. The first mill is an excellent example of the 

 centrifugal force and centrifugal couple, while the second lacks the 

 centrifugal couple, the instantaneous axis and the axis 

 of angular momentum being parallel. 



Let us calculate the couple involved in the con- 

 strained motion involved in a regular precession, as 

 here applicable, in terms of the constants of 81. If 

 the angular momentum make with the axis of figure 

 the angle a, its end is at the distance from the axis 

 of the fixed cone H sin (a + -9 1 ), so that it moves with 

 the velocity vH sin (a + #). This must be equal to the 

 applied couple, 



60) K=vH$m(a + &). 



Now resolving H parallel and perpendicular to the axis of figure, 

 we have 



61) H cos a = Ceo cos u, H sin a = AM sinw, 

 so that 



62) K= va(Asmucos& + 

 But we have, 81, 



-,o\ G> V 



lo) - = . j 



sin -9 1 sin u 



14) a? 2 = ^ -f v 2 - 



from which 



ca sin u = v sin #, & cos u = [i v cos #, 

 so that finally 



63) K = v{Av8w&eo8& 4- (7 sin -9- (^- 



It is to be noticed that the body will perform a regular precession 

 under no constraint or other applied couple, if putting .ZT=0, 



n C 



64) 



C-A 



9O. Heavy Symmetrical Top. We will now take up one of 

 the most interesting problems of the motion of a rigid body, namely 

 the motion of a body dynamically symmetrical about an axis, on 

 which its center of mass lies, and spinning about some other point 

 of that axis. This is the problem of the common top or gyroscope. 

 In order to determine the position of the top it will be convenient 

 to introduce three coordinate parameters, namely the three angles 

 of Euler. Let these be # the angle between the J^-axis, which 

 we take as the axis of symmetry, and the fixed vertical Z f -axis 



