474 Mr. H. W. Chapman on 



This gives 



i. p. 



*(*»+*&)(% +A+2C SH I ^+2AC*^ 

 + (?5 i ^+C-»«( OT +A + C- i )}>^-, 



_ We see from this that if /Lt andyu, =0, the top must fall if 

 w 3 = and A be positive. It is therefore necessary to give 

 the body an initial roll as well as an initial spin. The con- 

 dition becomes, supposing co 3 not zero, 



h _ / C A ghK 1 



i. «. / C A h „ ah 



Putting in the values for h, a, C above we get 



l-4ma> 3 — -375m 2 >75. 



This shows that to represent a real case we must have m 

 and ws rather large as the spins of tops usually go. Take 



?» = 10, g) 3 = 10 radians per second. 

 These give 



\=8'094, K=145, n 2 = 97-94. 

 And equation (16) gives 



65-2 V- 103-15/x-°- 143-73/*- 59-4 6 + 8'956(-375+/*) 

 • 2= x N/31 3'4-f96/>6-83^ 2 



** ~~ -7826 + -3261/*, 



This of course has a root /Lt = 0, and the next root is ^ = '328,. . . 

 which corresponds to a rise of the axis through about 19° 9'. 



On tracing the curve and integrating it by graphical 

 methods we get fig. 1 (PL XL) which gives the period of the 

 nutation. The integration was done by two methods, shown 

 in fig. 1 ; firstly, by inverse summation, i. e. the curve was 

 divided up by ordinates and the height of the mid-ordinates 

 set off along a vertical as MII = NP, a polar distance MO 

 was taken and pp drawn between the extreme ordinates 

 perpendicular to OIL It can be seen that the curve thus 

 obtained is such that the ordinate is proportional to t. The 



