the Problem of Columbus. 46U 



and 



^^V^^'^ 



i£ C 4 is negative. 



We see that in either case for the approximation ever to be 



CM " 

 valid we must have -£-j small, as is otherwise obvious, for 



C must be small. 



This ensures \f y — 2 ° 4 J being positive, which is also 



seen to be necessary. 



In the first case we see that u cannot remain small. This 

 therefore gives the motion near the origin, but need not be 

 further considered. 



In the second case we see that for the approximation to 

 hold always, both 



2(V 2C 4 



and - g- - ^?^A must be small? 



/. e. C and C 2 must both be small. 



This is really a case of the last article, the egg oscillating 

 between two cones both near the vertical. 



If C = Owe have 



and C 4 is negative, so 



Q 9 



i...*- ( -_V" __ -J 



This will be always valid near the vertical, and we see that 

 in this case the body oscillates through the vertical inside a 

 cone whose semi vertical angle is given by the next root. 



If C 2 is small the approximation is always valid. We see 

 by (17) that when C o = 0, i.e. 



A ^ sin 2 6 takes the form Vd* + . . . 



Phil Mag. S. 6. Vol. 5. No. 28. April 1903. 2 I 



