[ 458 ] 



XLV. The Problem of Columbus. By H. W. Chapmax. 

 B.Sc, University College, London*. 



[Plate XI.] 



I 



1. ■? an ordinary hard-boiled egg be laid on its side on a 

 horizontal plane and be then given a spin round the 

 vertical its axis will rise and at last stand up on its end, still 

 spinning ; in this position it will remain until its motion 

 fails and it falls down. This problem I term Columbus' 's 

 Problem. To a first approximation it would appear as if the 

 dynamical problem could be treated as identical with that of 

 a body symmetrical about an axis and with a hemispherical 

 end moving on a perfectly rough horizontal plane. 



The only references I have been able to find to the motion 

 of a body of this nature are first, in Jellett's ' Treatise on 

 Friction/ where the case is discussed in which the spin 

 round the long axis of the egg is taken so large that all the 

 other motions are small in comparison with it. A second 

 reference occurs in Eolith's 'Advanced Eigid Dynamics,' 

 Article 244, Example 4, where the equation of angular 

 momentum (my Equation 12) is given. 



There are several investigations of the more general problem 

 of a solid of any shape moving on a plane. In Poisson's 

 Traite de Mecanique the small oscillations of a body on a 

 smooth plane are treated of. In the 5th and 8th vols, of 

 CrehVs Journal M. Cournot considers the case of a body on 

 a perfectly rough and also on an imperfectly rough plane, 

 but he confines himself almost wholly (i.) to showing that 

 there are sufficient equations obtainable to determine the 

 coordinates of the body, and (ii.) to the consideration of 

 initial motions. 



It therefore appeared of interest to examine whether the 

 problem of the egg could not be completely solved in the 

 most general case. This solution is effected in the following- 

 pages. 



2. Let be the point of contact at any time, C the centre 

 of the hemispherical base, and Gr the centre of gravity 

 of the egg. Let OCZ be the vertical through 0, OX the 

 projection of CG, the axis of the body, on the horizontal 

 plane, and OY a perpendicular to OX in this plane. Let 

 00 = a, CG = /i ; let the moment of inertia of the body about 



* Communicated by Prof. Karl Pearson, F.P.S. 



