284 VII. DYNAMICS OF ROTATING BODIES. 



which is constant, so that the motion is a regular precession, without 

 rise and fall. There are thus, for a given velocity of spinning, and 

 a given angle of inclination with the vertical, two values of the 

 velocity of precession. We may also find these by considering the 

 equation 90), putting & constant in which gives, if sin^ is not zero, 



105) -4 



a quadratic for ty 1 with the roots 



These values are real if 5 2 > 2acos# i . If the top be spun so fast 

 that - a ? S is a small quantity whose square may be neglected, we 

 find for one value of ty* 



which is a large quantity of the order of 6, while the other root is 



which is a small quantity of the order of y Of these it is the 



slow precession which is usually observed. 



It is to be observed that if we put ^' = v, (p* = /A, the first of 

 equations 82) gives for P$ the same result as obtained for K in 63). 

 When we make a vanish, so that the body is under the action of 

 no impressed moment, the root ^ becomes zero, so that the axis of 



figure stands still, while the root ^ becomes - <r> that is, the body 



performs a Poinsot -motion around the vertical as the invariable axis. 

 Thus the effect of the impressed forces may be looked upon as a 

 small perturbation of the Poinsot -motion. 



We will now consider the motion when the condition 103) is 

 not fulfilled. From equation 78), we have t given by the elliptic 

 integral, 



109) t=C- =^= 



J ya(s-gj(g-sj(e-zj 



We may easily find two limits within which this value lies, by 

 substituting for the factor ]/# 3 in the integrand its greatest and 

 least values, as we did in the case of the spherical pendulum. 



