﻿Pseudo-Regular Precession. 191 



Then in this Steady Motion, 



7^ -y^T -fwi — 0, where 



m 2 OK 2 . . \ „ 2 AP 2 



n 2= A^^ 4c0 ^ = 7- 2cOS ^ + X = OATOP^ 



£-2£ OA 



m 2 ~AP 2 UA * 



This result is exact and reached without any approxima- 

 tion : and the slightest disturbance will give a nutation 

 Q = Q cos {mt + e) , beating m/27r times a second, and the 

 apsidal angle, from node to node, is 



Mr " 0P 



\ m AP 



In Darboux's representation of top-motion by a deformable 

 articulated hyperboloid of the generating lines, the model is 

 flattened into a rigid framework for Steady Motion ; and 

 KM, KN produced to double length at S, S' will make the 

 focal line SS' parallel to MN ; this will be revolved about 

 the vertical line ON with constant angular velocity jul. The 

 small nutation will be due to a slight play or baeklash in 

 the frame. 



13. The same argument can be applied to the invisible 

 oscillation of a Simple or Spherical Pendulum, or to the 

 apsidal angle of a particle describing a horizontal circle on a 

 smooth surface of revolution about a vertical axis. 



Taken as the axis Oy, the general equations of motion of 

 the particle are 



1 dx 2 Idy 2 1 „d^ 2 , „ , . 



and x 2 -j- = K (impulse) ; 



d v d*\l/* 



so that, with -=-' =Q, and eliminating -j- , 



and differentiating with respect to #, dx = Qdt, Q C -~ = ~, 



dt\ 1+ dx 2 ) + ^ dx dx 2 x s +9 dx~ V > 

 1 d^ ( djf\ dQdylij td 2 y\ 2 



Q dt 2 \ + dx 2 ) + dt dx dx 2 "*■ H \dx 2 ) 



~dy d 2 y , 3K 2 d 2 y A 

 dx dx z x* J dor 

 exact equations. 



