﻿180 Sir G. Greenhill 



on 



an impulse couple must be applied, given by a horizontal 

 tap of appropriate amount. 



The word moment or momentum is of such frequent 

 occurrence in dynamical theory that we prefer to replace it 

 by impulse instead of momentum, linear or rotational. 



Representing then the impulse (rotational) CR due to the 

 rotation R about the axle by the vector OC, the impulse 

 required to start the rotation from rest, or reversed to stop 

 it again, the axle OG in steady motion will move round the 

 vertical OG at a constant (horizontal) inclination, with pre- 

 cession /j,, such that the vector velocity of K, the end of the 

 resultant impulse OK, is equal to the impressed couple of 

 gravity ; then CR{A=gM.h, M.h denoting the preponderance 

 or first momeni about 0. 



This result, true accurately when the axle OC is hori- 

 zontal (fig. 1), is obtained at any other inclination 6 of the 

 axle with the vertical (nadir or zenith, figs. 2, 3) in 

 the elementary Kindergarten treatment, where the top is 

 supposed spinning so fast that the deviation is insensible 

 of axial impulse OC and resultant impulse OK, and then 

 the velocity of C may be equated to the gravity couple, 

 making CR//, sin 0=gM.h sin 6, as before, when the axle was 

 horizontal, on dividing out sin 6. 



2. Hanging down inert, vertically from in fig. 2, the top 

 forms a compound pendulum of S.E.P.L. (simple-equivalent- 

 pendulum-length) 0P = £ = A/M/t, A denoting the trans- 

 verse-diametral M.I. (moment of inertia) at 0, and as 

 above the axial M.I. ; and in small invisible oscillation in a 

 plane, the top will swing as a pendulum, and beat njir times 

 a second, where n 2 =g/l, An 2 =gMh, or make a swing or 

 beat in ir/n seconds ; MA may be called the preponderance 

 about 0. 



Falling down from rest from" the upward vertical position, 

 the top will have acquired the angular velocity co in the 

 lowest position, such that by the Energy-Principle, 



iAo) 2 = %M/>, 



and the equivalent rotational impulse Aa) = 2A??, while 

 An 2 =gMh is the equivalent of P in the Kreisel-Theorie. 



The impulse 2An = 2 ^(gMhA) is a dynamical constant of 

 the top, and to a geometrical scale may be represented by a 

 length Jc, in addition to the S.E.P.L. 0P = Z measured along 

 OC ; and then if in any assigned top motion the constant 

 impulse component G and CR, about the vertical OG and 



