112 GOVERNORS. 



If a is a negative quantity, this will indicate an oscillation the amplitude 

 of which continually decreases. If a is zero, the amplitude will remain constant, 

 and if a is positive, the amplitude will continually increase. 



One root of the equation (12) is evidently a real negative quantity. The 

 condition that the real part of the other roots should be negative is 



Y G 



<i uantit y- 



This ia the condition of stability of the motion. If it is not fulfilled there 

 will be a dancing motion of the governor, which will increase till it is as great 

 as the limits of motion of the governor. To ensure this stability, the value of 

 )" must be made sufficiently great, as compared with G, by placing the weight 

 H' in a viscous liquid if the viscosity of the lubricating materials at the axle 

 is not sufficient. 



To determine the value of F, put the break out of gear, and fix the 

 moveable wheel ; then, if V and V be the velocities when the driving-power 

 is P and P', 



P-P 



F= 



V-V" 



To determine G, let the governor act, and let y and y be the positions 

 oi' the break when the driving-power is P and P", then 



PP f 

 G = - A-. 



y-y 



General Theory of Chronometric Centrifugal Pieces. 



Sir W. Thomson's and M. Foucault's Governors. Let A be the moment of 

 inertia of a revolving apparatus, and 6 the angle of revolution. The equation 

 of motion is 



where L is the moment of the applied force round the axis. 



