280 . DYNAMICS OF ROTATING BODIES. 



corresponds the same value of ^, or the path of the point of the top 

 is the same, but described at a different rate. Thus the shape of 

 the path depends on the three ratios of the constants, or there is a 

 triple infinity of paths. As for the meanings of the constants, 

 a depends simply on the nature of the top, irrespective of the motion, 

 and by comparison with 80 is seen to be inversely proportional to 

 the square of the time in which the top would describe small oscilla- 

 tions as a pendulum, if supported with its apex downwards, without 

 spinning. If we change the top, we may obtain the same path by 

 suitably changing a, &, /3 as just described. These three constants 

 depend on the circumstances of the motion, b being proportional to 

 the angular momentum about the axis of figure, or to the velocity 

 of spinning, /3 to the angular momentum about the vertical, and 

 a depending on both the velocity of spinning and the energy constant. 

 Expressed in terms of the initial position and velocities they are 



a= 2^Z ^_C_ 



92) = 



With the convention that we have adopted, a is positive. As it is 

 evident that any path may be described in either direction, we shall 

 obtain all the paths if we spin the top always in the same direction. 

 We shall thus suppose b to be positive, while /3 may be positive or 



negative, according to the sign and magnitude of (-5?) and cos # . 



dz \*/o 



Since -=- is real, f(si) 78) must be positive throughout the 



motion, except when it vanishes. Since /"(I) = (j3 b) 2 and 

 f( 1) = (|3 -f b) 2 are both negative, f(oo) = oo and f( oo) = oo, 



the course of the function 

 f(z) is as shown in Fig. 93. 

 Thus it is evident that f(z) 

 'Z has three real roots, two 

 #!, 2 , lying between 1 and 

 - 1, while the third, # 3 , 

 lies outside of that interval 



on the positive side. Thus the motion is confined to that part of 

 the #-axis between $ if z 2 , and the apex of the top rises and falls 

 between the two values of <# whose cosines are ^ and # 2 . The 

 triple infinity of paths may be characterized by giving the three 

 roots all possible real values, instead of giving the constants ft, /3, a. 

 In practice it will be convenient to give the two roots indicating 

 the highest and lowest points reached by the apex, and the value 



