the Problem oj Columbus. 475 



other method was Proll's method ; in this ordinates are taken 



as before and from the top of the mid-ordinate a constant 



length QG is set off to the base-line; qq' is then drawn between 



the extreme ordinates perpendicular to QG. The time in this 



case is proportional to the arc of the curve. These two 



methods both give about *685 second as the complete period 



of nutation; the resulting curves are shown in PI. XI. fig. 1. 



In fig. 2 the precession is dealt with, \fr is found in terms of /-t, 



the equation being 



•3959(*375 + /*) 

 ♦6594 t (l^^) = 8-094^ V . 9794 ;. 3 ^. 25v ; 



(j, is found in terms of t from fig. 1, and thus a yjr and t curve is 

 constructed. This is integrated by the ordinary sum-curve 

 method, and the amount of precession during a period of the 

 nutation found. 



Fig. 3 is a polar diagram in which the ray is taken pro- 

 portional to sin 6 and the angle to yfr. It starts on the outer 

 circle at C l5 touches the inner at C 2 , and meets the outer again 

 at C, the period of nutation being a little more than that of 

 the precession. It is the projection on a horizontal plane of 

 the motion of the centre of gravity relative to the centre 

 of the hemisphere. 



The coefficient of friction necessary that the egg may begin 

 without slipping is easily found. Initially we have = 0, 

 £ = 0; so we have 



F 2 =0, 



yr = a6, — m 2 h — m <o.,a 



= — a/^o — "* 2 ^ — ma) 3 a, 

 R 

 M — h P6+9- 



Working out p from (18), we get ^ =*728. 



This shows that slipping would take place a:> w rule, but 

 that it is perfectly possible to get surfaces rough enough 

 to prevent it. We find /jl = 11*47; thus the /u, terms are 

 much less important than the others, so that slipping would 

 be less likely if m and &> 3 were smaller ; but this would 

 involve a larger radius for the hemisphere. 



14. The results obtained in this paper show : — (i.) That the 

 axis of an egg-shaped body would certainly not rise towards the 

 vertical unless, when its axis is horizontal, it receives not only 



