282 



VII. DYNAMICS OF ROTATING BODIES. 



If s denote the angle made by the tangent with the radius vector, 



d% dip dib 1 



96) 



tans 



1-** 



dip 

 dg dt 

 -z ds _ 

 r^z*~di ~ 



Accordingly for 1} 29 roots of f(z), tans = oo, and the axis cone is 

 tangent to the limiting cones , unless at the same time the numerator 

 vanishes. This can he the case for only one of the limiting values. 

 In case the numerator vanishes, say for = lt we have ft = b0 lt 



97) 



tans = 



which vanishes for = 19 so that the cone has cuspidal edges. If 

 the top is merely set spinning, and let go, so that -TT- = -^ = it 



Cv v Civ 



evidently begins to fall vertically, so that the cusps are on the upper 

 limiting circle, while the path is tangent to the lower. The reason 



Fig. 95. 



Fig. 95 a. 



Fig. 96. 



that the top starts to fall vertically is, of course, that the gyroscopic 



-J Q^ 



action does not begin until the velocity of falling -^ begins, as shown 



in 87). It is to be noticed that when z is positive, as in the ordinary 

 top, the horizontal projection has the cusps turned inward, while 



Fig. 96 a. 



Fig. 97. 



Fig. 9 7 a. 



when z is negative, as in the gyroscopic pendulum, the cusps are 

 turned outward. The three types of motion are shown for the first 



