the Problem of Columbus. 463 



(ii.) A<C and 7t>0. The condition is never satisfied if 



IT IT 



6 < -r- . If 6 > ~ the conditions o£ the problem are not 



_ — 



satisfied unless the body is spherical. In this case the top 

 behaves as in (i.), changing the signs of (A — C) and cos 6. 

 (iii.) A>C, 7i<0. The condition is satisfied as long as 



O<0 < 5- . If 6> it is as in the first case, changing the 



signs of cos 6 and It. 



(iv.) A<C, h<0. This is similar to the first case, if we 

 write C — A for A— C and | h | for h. 



In any case the position for which [ co s \ is a minimum, if 

 a possible one, does not depend on the initial motions. Also 

 <w 3 cannot become zero or change its sign, for the roots of 



-Yvuf -f Asin 2 + C{ — hcos#) =0 are clearly imaginary, the 



left-hand side being the sum of three positive quantities. 

 6. Substitute from (12) in (11) for yjr sin 2 <9 ; 



{A + & +cos *T M x - c *C + H } 2 



xx * sin- u 



L A 2 sin 2 4 A 



T (?M Mia VJ- a ^ L OTT +lJ 



• / A i A 2 , ~A A 

 ya w iVl a- a J 



<r A-siirV 



