MOTION OF A TOP 



313 



so that /(cos ) is negative. We easily find, from equation (143), 

 that 



so that /(I) is positive. 



Again, f(- 



so that /( 1) is positive, and 



/(+ oo) = 2 Mgh B (+ oo) 8 , 

 which is negative. Thus we have seen that 



when cos = + oo, /(cos 0) is ; 



when cos 0=1, /(cos 0) is + ; 



when cos 0= cos , /(cos 0) is ; 



when cos 0= 1, /(cos 0) is +. 



Thus the three roots of the cubic /(cos 0) = lie as follows : 



a root = l between cos = 1 and cos = cos ; 

 a root = 2 between cos = cos and cos = 1 ; 

 a root for which cos is numerically greater than unity, giving 

 no real value for 0. 



We see, therefore, that 

 the only points at which 



can vanish are 0=0j, 



CLL 



and = 2 . Moreover, at 



these points - - vanishes, 

 at 



and as there are not co- 

 incident roots at either 



. . d0 , 



point - changes sign on 

 at 



reaching these points, so that can range only between the values 

 B l and 2 . 



Thus the axis of the top oscillates between the two cones = l 

 and 0=0 2 . 



