the Problem of Columbus. 467 



so that x can only become zero after an infinite time and the 

 axis of the body approaches asymptotically to a certain cone. 

 If the root be of higher order still, 



and there is again asymptotic approach to a cone. 



(7) There is no relevant root. 



This is impossible, for we have the numerator in (17) posi- 

 tive in every possible position, otherwise /j, would become 

 imaginary. 



When fi = 1 it is 



--°& +*)•- » IA+ ('+■ )'} 

 ~N/A + (. 4 +0"-C-')-°y. 



When fj,= — 1 it i 



is 



-iVA^e-o'-e-o^}- 



These are both negative or zero. 



So that there is always a relevant root. 



We also see that the egg cannot stand up unless 



VA + (a+ i ) 2 -^e +i )=°- 



This is the condition found in art. 7, it will be observed that it 

 is independent of the value of fi . 



We see by this that if the top be started with fju between 

 two roots, it will oscillate up and down between the two cones 

 defined by these roots, or else approach one of them asym- 

 ptotically. 



If it be started at a single root we have yu, = 0, but /i will 



clearly not be 0, for this would involve I -j— ) = () ' ^ N is 



\dfi /n=ix 



the numerator in (17), and the root would be double. 



If it be started at a double root//, =0 and/A =0, so we have 



