Mr. J. C. Maxwell on Governors. 395 



The work spent on the liquid in unit of time is 



dfW dW' 



dt dt 

 Equating this to the work done, we obtain the equations of motion 



.A^+Bf+^Qg^™^ . . . . (7) 



These equations apply to a tube of given section throughout. 

 If the fluid is in open channels, the values of A and C will depend on 

 the depth to which the channels are filled at each point, and that of 

 k will depend on the depth at the overflow. 



In the governor described by Mr. C. W. Siemens in the paper 

 already referred to, the discharge is practically limited by the depth 

 of the fluid at the brim of the cup. 



The resultant force al the brim isf= V '# 2 -f-tuV"« 



If the brim is perfectly horizontal, the overflow will be propor- 

 tional to X2 (where x is the depth at the brim), and the mean square 

 of the velocity relative to the brim will be proportional to x, or to Qf . 



If the breadth of overflow at the surface is proportional to scm t where 

 x is the height above the lowest point of overflow, then Q will vary as 



x n+ i 3 and the mean square of the velocity of overflow relative to 



1 

 the cup as x or as n+%' 



Q 



If n = —\, then the overflow and the mean square of the velo- 

 city are both proportional to x. 



From the second equation we find for the mean square of velocity 



?=-K B " +0 ?) + "t-^+'>- ■■■<»> 



If the velocity of rotation and of overflow is constant, this becomes 



l^'l -*<*+')• • • OO) 



From the first equation, supposing, as in Mr. Siemens's construc- 

 tion, that cos a = and B = 0, we find 



...... L =<"' 2Q §- • • • • (») 



In Mr. Siemens's governor there is an arrangement by which a 

 fixed relation is established between L and z, 



whence 



L = -S*, (12) 



