﻿Pseudo-Regular Precession. 181 



the axle OC, is represented to the same scale by d and D', 



a, a' = G, cr = m' 



/t ' 2An ' a 



in Darboux's notation (a different use of A from that 

 employed above). 



Time can be reckoned in the pendulum beat, irjn seconds ; 

 and the relation, CB. /n=gM.h = An 2 , can be written 



jji _ A n 



n ~CR ; 



or expressed in words, the number of beats per circuit of the 

 axle is C/A times the number of revolutions of the top per 

 double beat. 



The resultant impulse vector being OK, the component 

 perpendicular to the axle, if horizontal, as in fig. 1, is 



A _ (?MAA _ A 2 n 2 

 ^"""CfT" CR' 



or to the geometrical scale, OC.CK = i& 2 , in the steady, 

 regular precession. 



For brevity we are allowed to assume tacitly the geo- 

 metrical scale, and to replace any dynamical quantity in an 

 equation by its vector length, such as the axial impulse CR 

 by the vector length OC, or a'. 



3. To change this steady motion of the axle into a 

 penultimate pseudo-regular precession, another impulse is 

 applied about a vertical axis, supplied by a horizontal tap 

 on the axle perpendicular to the plane OCK, in fig. 1. 



This will cause CK to grow to CK 3 , and the resultant 

 impulse to change from OK to 0K 3 ; and to make the 

 pseudo-regular precession advance through a series of cusps, 

 we find that KK 3 = CK, and the axle rises from OC to OC 2 

 at an angle 2 with the upward vertical, zenith ; where C 2 

 reaches the level GK 3 of K 3 ; and here # 3 , the inclination in 

 the lowest position, is \tt. 



By a general dynamical principle 



OK 3 2 -OK 2 2 =i* 2 (cos<9 2 -cos0 3 ) 



= 2OC.CK(cos0 2 -cos6> 3 ). 



For in the general unsteady motion of the axle of a top, 

 where the inclination 6 is varying, a new component KH is 

 added to the impulse OK perpendicular to the vertical plane 



