390 Royal Society : — 



fugal piece revolves on the principal axis, and is kept always at a 

 constant angle by an appendage which slides on the edge of a loose 

 wheel, B, which works on the same axis. The pressure on the edge 

 of this wheel would be proportional to the square of the velocity ; but 

 a constant portion of this pressure is taken off by a spring which acts 

 on the centrifugal piece. The force acting on B to turn it round 

 is therefore 



■piCtxl p X 

 d ' 



and if we remember that the velocity varies within very narrow limits 

 we may write the expression 



F 



(S-v 



where F is a new constant, and Y 1 is the lowest limit of velocity 

 within which the governor will act. 



Since this force necessarily acts on B in the positive direction, and 

 since it is necessary that the break should be taken off as well as put 

 on, a weight W is applied to B, tending to turn it in the negative 

 direction ; and, for a reason to be afterwards explained, this weight 

 is made to hang in a viscous liquid, so as to bring it to rest quickly. 



The equation of motion of B may then be written 



dt 



— F (S-\)- Y 2- w - • < 6 > 



where Y is a coefficient depending on the viscosity of the liquid 

 and on other resistances varying with the velocity, and W is the con- 

 stant weight. 



Integrating this equation with respect to t t we find 



B*-* (,-7,0 -Yy-W# (7) 



If B has come to rest, we have 

 r W\ Y 

 *=V 1+ F "J'+F* < 8 > 



or the position of the machine is affected by that of the governor, 

 but the final velocity is constant, and 



W 



Y.tTT-?' <») 



where V is the normal velocity. 



The equation of motion of the machine itself is 



M S= P ~ E - F (S- V ')- G ^ ...... do) 



This must be combined with equation (7) to determine the motion 

 of the whole apparatus. The solution is of the form 



^A^i'+A^+Ag^'+V*, (11) 



