92] 



MOTION OF APEX. 



281 



of -~ the horizontal angular velocity at one qf them, which three 



data completely characterize the motion. 



Since g is an elliptic function of the time ? the rise and fall is 

 periodic, and after a certain time, g will have attained the same 



value, and so will -^ and -A accordingly during successive periods 

 the angles ^ and <p will increase by the same amounts. The 

 horizontal projection of the axis of the top advances at the rate -jj' 

 This vanishes and changes sign if g = ~ and then we have 



The second factor is positive if - < 1, which will be the case if 



that is if the top is spun fast enough or (~J is small enough. We 

 must then have 



that is, 

 95) 



This will certainly be the case if (- is negative at the highest 



point of the path, and if it be positive and greater than -^ If the 



/? 



top be spinning so slowly that j- is greater than unity, g cannot 



attain this value, and ~ will never vanish. It is evident that when 

 at 



~ = the projection of the path on the horizontal plane has a 



radial tangent (unless g = 0). 



The axis of the top describes a 

 cone limited by the two circular cones 

 of angles ^ , # 2 , where g^ = cos & v 

 # 2 = cos #2 . It is in general tangent 

 to the two cones, as may be shown as 

 follows. The projection of the apex 

 on the horizontal plane X'Y 1 has the 

 polar coordinates (Fig. 94), 



Fig. 94. 



