﻿Pseudo- Regular Precession. 189 



This caa be applied by hammering the rim o£ the bicycle 

 wheel with a stick in a vertical blow at its highest point. 

 Then 



OK3 2 -OK 2 2 = OK3 2 -OC3 2 = C 3 K3' 2 



= 2 OM . MK (cos 2 -co8 3 ), 



M3K3 . K 3 D 3 = 20M . MK l -^ } M 3 K 3 . 00=20M . MK, 



and this determines the level of AM 3 in fig. 5, and provides 

 a geometrical construction for the position of 3 . 

 For 



0M 3 . M 3 C 3 = OA sec 3 . M 3 K 3 cos 3 = OA . AG, 



so that, if B is taken in OG where AB = OA, 



4 0A . AG = OG 2 -BG 2 = OK 2 2 -K 2 B 2 ; 



and the circle centre K 2 {in d radius K 2 B will cut off a length 

 OE on OK 2 = OF = 20M 3 , and so determine the direction of 

 OM 3 C 3 , the axle in the lowest position. 

 When OC is horizontal, as in fig. 1, 



cos# 3 = 5 CH 2 = 2OC.CKcos0 2 = 2CK 2 , KH = KC, 



as at first ; and OC = OM, M 3 K 3 = 2MK, 9 2 + 6, = tt, 

 the axle oscillating to an equal angle above and below the 

 horizontal. 



11. When the cusp motion is pseudo-regular and in small 

 loops, it can be projected on the tangent cone of a sphere in 

 a series of small hypocycloidal branches, and the motion is 

 realised as discussed in the Principia, Book I, section X, 

 when the tangent cone is developed into a plane and gravity 

 radiates from a centre. 



With the axle horizontal, a necklace of brachistochrone 

 cycloids is formed round the equator, with mean regular 

 precession fi, fluctuating in azimuth between and 2fi, with 

 azimuth interval 2^ = 2 Kk, tending as the rotation Rand 

 axial impulse CR is increased to ttcos0 2 , in a zone above 

 the equator of angular width ^it — # 2 , and area 2 7m 2 cos # 2 

 on a sphere of radius a, and the number of cycloids in the 

 necklace would be about 2sec# 2 . 



12. Even in the steadiest smoothest Regular Precession a 

 close scrutiny will reveal under the slightest disturbance an 

 almost invisible deviation from a perfect circular motion, in 

 the shape of a progressive motion of an apse line, realised 



