378 



Mr. J. C, Maxwell on Governors, 



[Mar. 5, 



shaft. They will neither increase nor diminish if there are no other terms 

 in the equations. 



To convert this apparatus into a governor, let us assume viscosities X 

 and Y in the motions of the^ main shaft and the centrifugal piece, and a 



resistance applied to the main shaft. Putting ^ w = K, the equations 



become ^ (PQ dd , , ^ t 



B 



de 

 d^ 



dt 



dt 



■K- 



:0. 



(10) 



dt^ dt " dt 



The condition of stabiHty of the motion indicated by these equations is 

 that all the possible roots, or parts of roots, of the cubic equation 



AB7^^ + (AY+BXX + (XY + K^)?^-f GK=0 .... (11) 

 shall be negative ; and this condition is 



g+l) (XY+K^)>GK (12) 



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

 plied 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 calculation of the general equa- 

 tions of motion of these pieces, we may confine ourselves to the case of 

 small disturbances, and write the equations 



d'^d 



dB 



■ d(j) 



=0, 



= P-R, 



de'^ dt dt 



CS+Z^-T, 



=0, 



(13) 



de ' "dt 



where 9, 0, \b are the angles of disturbance of the main shaft, the centri- 

 fugal 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 for- 

 dK 



merly denoted by -^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 m+Y 



^ G — -T Cn^ + Zn 



,^ ,/X Y , Z\ gfXYZ /A, B_^C\ ^ Kn 

 "+HA+B + c) + "LABc(x+Y + zj I 



\ KTZ , KTJ ^ 

 j + ^ABC + ABC=^- J 



or 



= 0, 



1 



. 2/XYZ + KTC 



ABC 



(14) 



(15) 



