GOVERNORS. 



11;; 



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

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



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 <f>, then the force tending to diminish (f>, arising from the action of 



dP 

 gravity, springs, &c., will be -jj . 



The whole energy, kinetic and potential, is 



, ............. (2). 



Differentiating with respect to t t we find 



d^LdAdfT dBdj* dP\ d0tW d^_d^- 

 dt \*^ dt + *d<l>& + cfy) 4 l Tt dt 3 4 dt dt> 



T dQ_M/dAd0d$ A 

 dt ~ dt \ddt dt + dt" 



whence we have, by eliminating L, 



*/^\ i^^fo-X^-^ 



dt ( dt) "* 34 dt + * (ty dt d<j> " 



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

 increase <J>, depending on the squares of the velocities of the main shaft and 

 of the centrifugal piece. The fca-ce indicated by these terms may be called the 

 centrifugal force. 



If the apparatus is so arranged that 



P = ^o) 2 + const ............................... (5), 



where w is a constant velocity, the equation becomes 

 d 



VOL. II. 



15 



