288 



VII. DYNAMICS OF ROTATING BODIES. 



distance traversed in one revolution, we see that when the top is 

 spun rapidly, so that the precession is slow, the rise and fall is very 



/ c*\ 



rapid (for b = r -j-j ? and very small. For this reason it is seldom 



noticed, and this accounts for the popular opinion, expressed in many 

 text hooks, that the motion of a top is such that its axis describes 

 a circular cone with a constant angular velocity, or a regular 

 precession. Thus the reason of the vertical force of gravity producing 

 a horizontal motion remains a paradox. We have seen that such a 

 motion is the very particular exception, and not the rule, being only 

 exhibited when the necessary horizontal velocity is imparted at the 

 outset, so that the action of gravity is always balanced by the 

 centrifugal couple generated by the precession. If the necessary 

 velocity is not imparted, the top immediately begins to fall in 



Fig. 101 a. 



Fig. 101 b. 



obedience to gravity. The motion which we have just described is 

 called by Klein and Sommerfeld a pseudo- regular precession, and may 



be called a small oscillation about 

 a regular precession. In Fig. 101 

 are shown curves of the actual 

 path obtained by photographing 

 a small incandescent lamp attached 

 rig. 101 c. to the axis of a gyroscope, with 



0- nearly a right angle. 



94. Small Oscillations about the Vertical. In the discussion 

 which has just been given, it has been supposed that 1 # 2 was 

 not a small quantity. If however in the course of the motion the 

 axis of the top becomes nearly vertical this will no longer be true, 

 so that for this case a special investigation is necessary. Let 

 us suppose that & and #' are so small that in the kinetic potential 

 all their powers above the second may be neglected. Let us use for 

 coordinates the rectangular coordinates of the projection of the apex 

 on the horizontal plane, 



x = r cos ^', y = r sin ^, r = sin #. 



Using then the expression of 85) for the kinetic potential, with 

 W= Mglcosft, 



