MOTION OF A TOP 315 



when cos is a maximum, i.e. when the top is started vertical. 

 In this case the top will spin if 



2 MgJiB sin 2 3 

 > .4 2 (l-cos0 3 )' 



or if Qa> 



256. In general, for a top started vertically and with no velocity 

 except one of pure rotation about its axis, we find, on putting 

 cos = 1 in equation (144), 



/ (cos 0) = (1 cos 0) 2 [A*W - 2 MghB (I + cos (9)]. 



The roots of the equation /(cos 6) = are 



22 

 cos0=+l, +1, 



iMghB 

 Let us write il; = - ^ 



B 



2 MghB 



then when H 2 = fl 2 , the roots are 



cos0 = + l, +1, +1. 



When H 2 > H 2 the third root is greater than unity, and when 

 fl 2 < fl 2 the third root is less than unity, say cos 6 = cos , 

 where is a real angle, given by 



y-X -O- \M (! ^"^ * 



1=1 -- < 147 > 



Thus, as long as H 2 > H 2 the oscillations are confined within 

 the coincident limits = and 6 = 0, so that the top remains 

 vertical, but as soon as we have H 2 < H 2 the oscillations are 

 between the limits = and = . 



Suppose we start a top with angular velocity fl greater than H , 

 so that at first its axis is vertical and the only motion of the top 

 is one of rotation about its axis. Then the real roots for are 

 0, ; there is therefore no range of oscillations, and the axis of 



