392 



Royal Society : — 



is a function of d>, then the force tending to diminish (p, arising from 



dP 

 the action of gravity, springs, &c, will be — . 



The whole energy, kinetic and potential, is 



E =2 A ^ 





+ P =fuw. 



Differentiating with respect to t, we find 



efy/l dA ddf.l clB "gl 2 cZP\ " 6 

 c^\2 cty cfcj + 2 d<p dt d<p) + < 





B 



^ d* 2 



~ dt~~dt\dd> dt at* dey . 



whence we have, by eliminating L, 

 M dtj~2~d^~di\ + 2d(j> It 



dF 

 d(p 



(2) 



(3) 



(4) 



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

 ing 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 maybe called the centrifugal force. 



If the apparatus is so arranged that 



P=^ Aw 2 + const., 



where io is a constant velocity, the equation becomes 



dt\ dt)~2 d(j>\dt\ 



,\ 1 dB dip 

 W ) + 2~a^dt 



(5) 



(6) 



In this case the value of (p 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 d and <p may be 

 written 



d 2 , dA d(f>_ T 



df ^ dd) dt ' 



^d 2 $ dA dO n 

 dt d<p dt 



(8) 



dA 



The period of such small disturbances is —r- (AB)~£ revolutions 



of the 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 viscosi- 

 ties X and Y in the motions of the main shaft and the centrifugal 



