282 



Mr. J. C. Maxwell on Governors, 



[Mar. 5, 



the equation becomes 



(s- g)^ + (h-s»^,§ g)s,=o ; (5) 



and since ll and It] are independent, the coefficient of each must be zero. 

 If we now pnt 



2(^/)=L, 'Z(mpq) = W, S(»25^)=N, . . .„ . (6) 



where 



p'^Pr-^P.^+P^, m=2M^-^M2-^Pzq^^ and + q^^ + q,\ 



the equations of motion will be 



(8) 



If the apparatus is so arranged that ^ = 0, then the two motions will be 

 independent of each other ; and the motions indicated by ^ and t] 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 4> that of the differential system on the governor ; then the equation 

 of motion becomes 



eae+*a^+ (g-if -Mg)s^ + (H-Mg-Ng)a,=o. (9) 



(10) 



and if 



l7] = m6 + ^l(l>,J 



and if we put 



L' =LP' +2MPR -f NR' 



M'=LPQ+M(PS +QR) + NIIS, ]^ . . . (11) 

 N'=LQ^ +2MQS +NS^ 

 the equations of motion in and <p will be 



e+p;ir+QH=L'^+M'f^, 



dt at' 

 $ + r;5^+Sh=M' ^ +N' ^ 



(12) 



dt^ df 



If M' = 0, then the motions in Q and^ will be independent of each other. 

 If M is also 0, then we have the relation 



LPQ+NRS^O; (13) 



and if this is fulfilled, the disturbances of the motion in 6) will have no effect 

 on the motion in 0. The teeth of the differential system in gear with the 

 main shaft and , the governor respectively will then correspond to the 

 centres of percussion and rotation of a simple body, and this relation will 

 be mutual. 



