394 Royal Society : — 



I have not succeeded in determining completely the conditions of 

 stability of the motion from this equation ; but I have found two 

 necessary conditions, which are in fact the conditions of stability of 

 the two governors taken separately. If we write the equation 



n 5 +pn* + qn 3 + m* + sn + t, (16) 



then, in order that the possible parts of all the roots shall be nega- 

 tive, it is necessary that 



pq>*r and ps>t. (17) 



I am not able to show that these conditions are sufficient. This 

 compound governor has been constructed and used. 



On the Motion of a Liquid in a Tube revolving about a 

 Vertical Axis. 



Mr. C. W. Siemens' s Liquid Governor. — -Let p be the density of the 

 fluid, h the section of the tube at a point whose distance from the 

 origin measured along the tube is s, r, 0, z the coordinates of this 

 point referred to axes fixed with respect to the tube, Q the volume 

 of liquid which passes through any section in unit of time. Also 

 let the following integrals, taken over the whole tube, be 



f P kr 2 ds=A, f P r 2 dd=B, fphis=C, ..... (1) 



the lower end of the tube being in the axis of motion. 



Let (j) be the angle of position of the tube about the vertical axis, 

 then the moment of momentum of the liquid in the tube is 



h=a|+bq (2) 



The moment of momentum of the liquid thrown out of the tube in 

 unit of time is 



dH' 2r .dd> r 



df =pr ~ q dt + n Q ' cosa ' (3) 



where r is the radius at the orifice, k its section, and a the angle be- 

 tween the direction of the tube there and the direction of motion. 

 The energy of motion of the fluid in the tube is 



The energy of the fluid which escapes in unit of time is 

 5= W Q(A+z) + i^Q|'+^co.„Q^+j£Q.. . . . (5) 



The work done by the prime mover in turning the shaft in unit 

 of time is 



dt dt \d£ + dt )' •••..... (6) 



