﻿190 Sir G. G-reenhill on 



easily with the thread and plummet; utilised by Newton to 

 illustrate the Evection of the Moon, 



Returning to the general unsteady motion of a top in § 3, 

 and its vector impulse OH, the velocity of H imparted by 

 the gravity couple An 2 sin is horizontal and along KH 

 perpendicular to the plane GOC. This velocity is the rate 

 of growth of KH, added to the velocity of K carried round 



by the plane GOC with angular velocity ~- ; so that, with 



KO = A sin 6 -^, and putting -,, — Q, 



An 2 sin0 = A-^ + GK-^ =A-^+ . a , 



dt 2 at at A sin 6 3 



. xl _ OC-OGcos0 rrn OG-OCcos<9 



with GK = ; — z , KC = — — -. — ^ , 



sm 6 sin 6 



, dGK KC dKG GK 



and 



dd "sinfl' dd sin0' 



obvious geometrically on fig. 3. 



Differentiating with respect to 0, with d0 = Qdt, 



1 d 2 Q 2 a GK 2 -fKC 2 ~GK.KCcos0 



= n? cos 



e-~ 



Q dt 2 ~ A 2 sin 2 



GC 2 -3GK.KCcos<9 



cos# 



A 2 sin 2 6 



-( 



2 , 30M . ON\ a OK 5 



n 2 H r^ I COS #■ 



A 2 



an exact equation. 



In a state of perfect Steady Motion of Regular Precession, 



Q and -7- are zero, and, in dynamical units, from § 5, 

 OM.ON-AV, °M«i, gf-^-T' 



An />6 O.N /Z 2 Z' 



where X is the height of the equivalent conical pendulum, 



X=^ 2 = OA, l = g/n 2 = OY ; 



OC n u 



r-r— = - + - COS 6. 



An fM n 



OG 

 An 



n A a 



= - COS04- ~, 

 A 4 - W. 



GO 2 



OK 2 sin 2 (9 



An 2 



A 2 n 2 



in 2 6 /n 2 _ . , /j?\ . 9/i 

 - T — = 2 4-2cos0+^ sm 2 0. 

 ?r \/r nV 



