﻿184 Sir G. Greenhill on 



pendulum, with h', CR and ~d' zero, in which the axle is 



projected horizontally, with angular velocity 2 h = sin 2 $-—, 



and sinks down to an angle 3 with the nadir, rising up 

 again to the horizontal, and this makes 



}i—n V{i£ . sec 6 3 — cos Z ). 



The motion can be shown with a plummet on a thread, say 

 about 10 inches long, to beat as a pendulum twice a second, 

 a double beat period of one second ; whirled round swiftly, 

 the thread rising to the horizontal position, and sinking down 

 again periodically. 



Then we find 



v 2 2 = U 2 h 2 = 2gl(sec #3— cos 3 ), v% 2 = 2gl sec S , 



and in the conical pendulum, at angle # 3 , v 2 = ^v 2 2 . 

 The apsidal angle is found to be 



^ = l7T+K i /(l-2« 2 ) -*7T(l-3 COS 2 8 ) 



as the plummet is whirled round faster. 



5. But next suppose the axle OC is held at an angle 6 

 with the zenith, the wheel spun with impulse CR = OC, and 

 then released, in fig. 3. 



The axle will start from a cusp, at 6 = 6 2 , and the motion 

 in general is not expressible in finite terms as pseudo-elliptic ; 

 but it will represent a gravity brachistochrone on a sphere. 



To make the axle move steadily at the inclination 6 with 

 constant precession /*, the impulse vector CK is applied 

 perpendicular to the axle, such that yu, sin 6 being the com- 

 ponent rotation of the wheel about the axis OK' perpen- 

 dicular to OC, CK = A/isin# (the inertia of the stalk being 

 ignored), MO = Ayu,cos#, MK = A/*, KM drawn vertical to 

 meet OC in M, with the condition 



GK . //,= gravity couple =gMh sin 6 — An 2 sin 0, 



0M= GK : Arf 

 sin 8 /jl 



then the geometrical relation CM-f MO = OC becomes 



A?? 2 

 Ayucostf-f — =CR, 

 A 6 

 the condition for steady motion. Also 



OM.MC=AVcos<9 = fPcos<9, OM.MK==JP, 



so that K lies on this hyperbola with asymptotes OC, OG. 



