GOVERNORS. HI 



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 



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

 resistances varying with the velocity, and W is the constant weight. 



Integrating this equation with respect to t, we find 



Yy-Wt ........................ (7). 



If B has come to rest, we have 



f .............................. (8), 



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

 final velocity is constant, and 



W 

 V> + ^ = V. ................................... (9), 



where V is the normal velocity. 



The equation of motion of the machine itself is 



0, ............... (10). 



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

 whole apparatus. The solution is of the form 



x = A 1 e** t + A 1 e'* + A t e n * t + Vt ........................ (11), 



where n lt n n n, are the roots of the cubic equation 



MBn t + (lY+FB)n' + FYn + FG = Q .................. (12). 



If n be a pair of roots of this equation of the form aJlb, then the 

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



