I i 4 GOVERNORS. 



In this case the value of <j> cannot remain constant unless the angular 

 velocity is equal to to. 



A abaft with a centrifugal piece arranged on this principle has only one 

 velocity of rotation without disturbance. If there be a small disturbance, the 

 equations for the disturbance and <f> may be written 



d>B dA < , 



dA d$_ 



- 



dA 

 The period of such small disturbances is -/v (AB)~* revolutions of the shaft. 



They will neither increase nor diminish if there are no other terms in the 

 equations. 



To convert this apparatus into a governor, let us assume viscosities X 

 and Y in the motions of the main shaft and the centrifugal piece, and a 



dA 

 resistance G<f> applied to the main shaft. Putting -,, <a = K, the equations be- 



come 



The condition of stability of the motion indicated by these equations is 

 that all the possible roots, or parts of roots, of the cubic equation 



ABn t + (AY+SX)n t +(XY+K t )n + GK=0 ............... (11) 



shall be negative ; and this condition is 



(12). 



Combination of Governors. If the break of Thomson's governor is applied 

 to a moveable wheel, as in Jenkins governor, and if this wheel works a 

 steam-valve, or a more powerful break, we have to consider the motion of 

 three pieces. Without entering into the calculation of the general equations of 





