286 



VII. DYNAMICS OF ROTATING BODIES. 



Thus we see that by making 1) large enough we may make # 3 as 

 large as we please, when g l and # 2 are given, so that the approxi- 

 mation is better the faster the top spins. 



Let us now consider the horizontal motion, or precession. We have 



117) 



~dt 



We have already supposed g to be a small quantity, so that if we 

 neglect the square of j - ^ we have, after developing the second 



V 1 ~ Z Q ) 



factor of the denominator, 



118) 



l-fefc, 



dt 



(1-V 



Now inserting the value of g from 115) and integrating, 



Thus we see that ifj varies with a harmonic oscillation about the value 

 that it would have in the regular precession at the mean height , 

 of the same period as the vertical oscillation. If we project the 

 motion of the apex on the tangent plane to the sphere on which it 

 moves, calling f; the horizontal coordinate, and tj the distance moved 

 from the horizontal mean axis, we have, Fig. 99, 



120) 



yi^v 



Thus we see that the second terms of 115) 

 and 119) represent an elliptic harmonic motion 

 of the apex of the top. This is termed nutation. 

 We thus have a complete description of the 

 motion of a top when differing by a small 

 amount from a regular precession, as a regular 

 precession combined with a nutation in an 

 ellipse about the point which advances with 

 the regular precession. 



We shall now make an additional supposi : 

 tion with regard to the constants of the motion. 



We have seen from 108) that in the case of regular precession with 

 rapid spinning, the precession was slow. Let us then suppose that 



121) '=% P 



" 



is a small quantity of the same order as c, so that their squares and 

 product may be neglected. Since # is the cosine of the angle between 



