Mr. J. G. Maxwell on Governors. 391 



where n^ n 2 , n. A are the roots of the cubic equation 



MBrc 3 +(MY + FB> 2 + FYrc + FG=0 (12) 



If n be a pair of roots of this equation of the form a + V — lb, 

 then the part of x corresponding to these roots will be of the form 

 e at cos (bt + p). 



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

 plitude of which continually decreases. If a is zero, the amplitude 

 will remain constant, and if a is positive, the amplitude will conti- 

 nually increase. 



One root of the equation (12) is evidently a real negative quan- 

 tity. The condition that the real part of the other roots should be 

 negative is 



/F , Y\Y G 



\M BJ B~~ B = a P 0Sltlve quantity. 



This is the condition of stability of the motion . If it is not ful- 

 filled there will be a dancing motion of the governor, which will in- 

 crease till it is as great as the limits of motion of the governor. To 

 ensure this stability, the value of Y must be made sufficiently great, 

 as compared with G, by placing the weight "W in a viscous liquid 

 if the viscosity of the lubricating materials at the axle is not suf- 

 ficient. 



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', 



V-V 

 To determine G, let the governor act, and let y and y be the po- 

 sitions of the break when the driving-power is P and P', then 



y-y 



General Theory of Chronometric Centrifugal Pieces. 



Sir TV. Thomson's and M. FoucauWs Governors. — Let A be the 

 moment of inertia of a revolving apparatus, and the angle of re- 

 volution. The equation of motion is 



sO&H <» 



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



Now, let A be a function of another variable (the divergence 

 of the centrifugal piece), and let the kinetic energy of the whole be 



2 K dt\ + 2 h dt\> 



where B may also be a function of <£, if the centrifugal piece is com- 

 plex. 



If we also assume that P, the potential energy of the apparatus, 



2D2 



