Mr. J. C. Maxwell on Governors, 



397 



where S and H are the forces tending to increase £ and 77 respect- 

 ive^, no force being supposed to be applied at any other point. 

 Now putting 



&r= j p 1 Si;+2' 1 8i7, (3) 



and 



d 2 x _ d%\ d**} 



(4) 



the equation becomes 



(s-Xmp^-Zmpf^yU^-Zmn^-Zmr^y^O; (5) 



and since d£ and 5ij are independent, the coefficient of each must be 

 zero. 



If we now put 



S(wjp 2 )=L, S(«wpj) = M, 20<2 2 )=N, ... (6) 

 where 



i? 2 =iV+lV+l>3 2 > M=2Wi+M 2 +P 3 q3> and r =?i 2 +? 2 2 +? 3 2 > 

 the equations of motion will be 



(7) 



*-l*|+m|5», 



A> ■-*" (8) 



If the apparatus is so arranged that M = 0, then the two motions 

 will be independent of each other ; and the motions indicated by £ 

 and r] will be about conjugate axes — that is, about axes such that 

 the rotation round one of them does not tend to produce a force about 

 the other. 



Now let be the driving-power of the shaft on the differential 

 system, and $ that of the differential system on the governor ; then 

 the equation of motion becomes 



6*6 + ^+ (^-Lg-Mg)^ + (H-Mg-Ng)^ = ; (9) 





and if 



S£=P30 + 



and if we put 



1/ =LP 2 + 2MPR + NR 2 , 



M' = LPQ + M(PS + QR) + NRS, 



N'=LQ 2 +2MQS +NS 2 , 



the equations of motion in and (p will be 



e+P#+QH=L'g?+M'<S, 



at dt~ 



* + R# + SH = M' ~ +N' d ~t, 

 dfr at- 



(10) 



(11) 



(12) 



