280 



Mr. J. C. Maxwell on Governors, 



[Mar. 5, 



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



^S+«f+''^'QS+''i-«Q^=^. (^) 



«Jt + cf+i|.QH,.(A+.)-i,,.=f=0 (8) 



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 /= s/ g'^ -\- o)^r\ 



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



(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 a?^, where x is 

 the height above the lowest point of overflow, then Q will vary as a?^"^^, and 

 the mean square of the velocity of overflow relative to the cup as x or as 



1 



If 11 = — 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 



2/ d'<t> dq\ ^ dj^ cy - /n^ 



dt)^'-dtr^^^^-^'^' — • (9) 



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



|=4f_2,(, + .) (10) 



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

 cos a = and B = 0, we find 



^=9^'^ft (") 



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

 lation is established between L and z, 



L = -S^, (12) 



whence 



l=4¥-^^+¥-€ (-) 



If the conditions of overflow can be so arranged that the mean square of 

 the velocity, represented by is proportional to Q, and if the strength of 



