290 Mr Pocklington, On the Kinetic Theory of Matter. 



such that when the axis is turned through an angle <p from the 

 position for straight motion, there is a couple tending to move it 

 back equal to bsm<p. Let the distance of the steering axis 

 from the common geometrical axis of the other two wheels be a, 

 and let the mass of the added system be M. Let v be the velocity 

 of the steering axis, then the acceleration of the axis is v 2 / p normal 

 and dv/dt tangential to the steering wheel. The reactions ex- 

 ercised on the framework of the tricycle are equal and opposite to 

 M times these accelerations, and, since the mass of the framework 

 is zero, they must keep equilibrium with the couple b sin <f> 

 exercised by the springs on the framework and the reactions at 

 the wheels. The condition that this is so is found by the principle 

 of virtual work to be Mdv/dt = b sin <p, sin <j>/a = (b/a) sin 2 <£. Hence 

 unless cp is constantly zero, v increases, and as it cannot increase 

 without limit, the motion is one that tends to a final state given 1 

 by <p = 0. Excepting in special cases, absence of rotation of the 

 steering axis implies absence of motion in the system mounted on 

 it, and hence the final state is one in which the whole of the 

 energy has passed into energy of translation. In the final state 

 v is positive, and the triangle moves with its steering wheel 

 foremost. 



10. The case where the couple tending to restore the steering 

 axis to its normal position is b sin n(f> is interesting. There are n. 

 stable positions of equilibrium of the axis. On investigating the 

 small oscillations about the corresponding states of steady motion 

 in the case where the added body is a simple flywheel, those 

 steady motions are found to be stable in which the steering wheel 

 is foremost, the vibrations being then affected with decrement. 

 The general motion cannot be investigated, but we may perhaps 

 assume that, however started, the system will be ultimately found 

 to be moving in one of the stable steady motions. 



An experiment on the laboratory scale shows all the phenomena 

 mentioned above as well as might be expected. It is especially 

 interesting with an arrangement that approximates to that con- 

 sidered in § 9 to notice how, when the system is started with the 

 steering wheel hindmost, the oscillations of the steering wheel 

 increase till they absorb all the energy, and the tricycle comes to 

 rest. It then starts in the opposite direction, the oscillations die 

 out, and the theoretical final state of rectilinear motion is closely 

 approximated to. 



11. In the case of § 9, the final state is definite only in that 

 (f> has a definite value ; the tricycle is moving along an arbitrary 

 line. It is possible however to have a perfectly definite final 

 state. Let us add a massless arm to the front of the steering axis 



1 Hence in general the equation giving the final state of a system will contain 

 both velocities and coordinates. 



