GOVEBNORS, 117 



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



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 at the brim is /= -Jg* + wV. 



If the brim is perfectly horizontal, the overflow will be proportional to a;* 

 (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 Q*. 



If the breadth of overflow at the surface is proportional to x n , where x is the 

 height above the lowest point of overflow, then Q will Vary as x n+ *, and the mean 



square of the velocity of overflow relative to the cup as a; or as -^^ . 



v 



If n = -J, then the overflow and the mean square of the velocity are both 

 proportional to x. 



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



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



'-2g(h + z) (10), 



From the first equation, supposing, as in Mr Siemens's construction, that 

 cosa=0 and .6 = 0, we find 



