Mr. J. C. Maxwell on Gov 



ernors. 



393 



piece, and a resistance G<jf> applied to the main shaft. Putting 

 —T w = Ji, the equations become 





do 



+ x ^+ k ^+^=l, 



dt 



dr$ v ^0 ^efa 

 D dt 2 + * dt~^dt 



0. 



(9) 

 (10) 



The condition of stability of the motion indicated by these equa- 

 tions is that all the possible roots, or parts of roots, of the cubic 

 equation 



ABrc 3 + (AY+BX> 2 + (XY + K> + GK==0 . (n) 



shall be negative ; and this condition is 

 X . Y^ 



(a + b)( xy+k2 > gk - 



(12) 



Combination of Governors. — If the break of Thomson's governor 

 is applied to a moveable wheel, as in Jenkin's 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 cal- 

 culation of the general equations of motion of these pieces, we may 

 confine ourselves to the case of small disturbances, and write the 

 equations 



d 2 Q . ^dd . -rrdcp 



dt 2 



de 



+ X-j-+K-£ + T0 + Jty=P-R, 



dt 

 .d$ 

 dt' 



K 



d°1> , „dyfr 



dt' 



+ Z ^- T ^ 



dt 



= 0, 

 =0, 



(13) 



where 0, 0, $ 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 



dpL \ 



what was formerly denoted by -j- 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 2 + Xn 



K»+T 



J 



-K 



Bn+Y 











-T 



C?i 2 + Zn 



=o, 



or 



, 4 /X T Z\ s rXYZ/A,B J C\ , K 2 "| 

 + HA + B + cJ +f lABc(x + Y + zj + AbJ 



< 



XYZ + KTC + K 2 Z 

 ABC 



)+» 



KTZ KTJ f 



ABC + ABC °* J 



(14) 



(15) 



