1868.] 



Mr. J. C. Maxwell on Governors, 



277 



Now, let A be a function of another variable (the divergence of the 

 centrifugal piece), and let tlie kinetic energy of the whole be 



1 .^f l^f 



where B may also be a function of ^, if the centrifugal piece is complex. 



If we also assume that P, the potential energy of the apparatus, is a 

 function of ^, then the force tending to diminish (p, arising from the action 



dP 



of gravity, springs, &c., will be ^ . 



The whole energy, kinetic and potential, is 



-|-F=/Lc?a (2) 



Differentiating with respect to t, we find 



d(pn dX do 

 dt\2 1^ di 



4.1 ^ 5| _Lf^\ I A^^ — 4- -R^ ^ 



"^2 dt\ d<p)^ dt de- dt de I 



~ di~dt[dcj) dtdt'^ dej' 



whence we have, by eliminating L, 



(3) 



\ dtj~2 ~d6 'di 2 d(j> 'di\ d(j> 



dt{^dtj'~2 dcp dt ' 2 d(p dt\ d(p 

 The first two terms on the right-hand side indicate a force tending to 

 increase (p, depending on the squares of the velocities of the main shaft and 

 of the centrifugal piece. The force indicated by these terms miay be called 

 the centrifugal force. 



If the apparatus is so arranged that 



P=^ Aw^ + const., (5) 



where w is a constant velocity, the equation becomes 



dt\ dt) ~2 d(]) \dt\ )^2 dcj) dt 



(6) 



In this case the value of cannot remain constant unless the angular 

 velocity is equal to w. 



A shaft with a centrifugal piece arranged on this principle has only one 

 velocity of rotation without disturbance. If there be a small disturbance, 

 the equations for the disturbances and (j> may be written 



-S-%4=-' (^) 



«3-w^?r- • («) 



dA. 



The period of such small disturbances is -j- (AB)-^^ revolutions of the 



