GOVERNORS. 



115 



motion of these pieces, we may confine ourselves to the case of small distur- 

 bances, and write the equations 



dW 



dt 



64 

 lli 



dt 



~dl 



= 



.(13), 



where 6, <f>, $ are the angles of disturbance of the main shaft, the centrifugal 

 arm, and the moveable wheel respectively, A, B, C their moments of inertia, 

 X, Y, Z the viscosity of their connexions, K is what was formerly denoted by 



dA 



j-f- w, and T and J are the powers of Thomson's and Jenkin's breaks respectively. 



The resulting equation in n is of the form 



An' + Xn Kn+T J 

 -K Bn+Y 



-T Cri> 



= 



(14), 



or 



(15). 



I have not succeeded in determining completely the conditions of stability 

 of the motion from this equation ; but I have found two necessary conditions, 

 which are in fact the conditions of stability of the two governors taken 

 separately. If we write the equation 



= ..................... (16), 



then, in order that the possible parts of all the roots shall be negative, it is 

 necessary that 



pq>r and ps>t ............................ (17). 



I am not able to shew that these conditions are sufficient. This compound 

 governor has been constructed and used. 



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