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660 ME. C. W. SIEMENS ON UNIFOEM ROTATION. 



apparatus it has, moreover, been found that the conical pendulum is apt to fall into ellip- 

 tical rotation, whereby the subdivisions of time below the half second become inaccurate. 



Foucault's Governor. — Monsieur Foucault, of Paris, has treated the same question in 

 a different manner, substituting for the friction between solids, or of solids against fluids 

 which we employed to fix the angle of rotation of the conical pendulum, the varying 

 resistance of an air-propeller, according to the amount of area for the escape of air, 

 which area he regulates by means of the angular elevation of his conical pendulum. 

 The conical pendulum he employs is mounted in the manner of a Watt's governor of 

 the best construction, which renders his apparatus portable, and therefore convenient for 

 general purposes, while, on the other hand, it appears to depend for its correct action 

 upon the perfect condition of many mechanical details. 



Some months since an idea suggested itself to me which, while it furnishes the ele- 

 ments of a very general and complete solution of the problem under consideration, 

 appears to possess also a separate scientific interest, which chiefly induces me to bring 

 the subject before the Royal Society. 



Liquid in rotation. — If an open cylindrical glass vessel or tumbler containing some 

 liquid be made to rotate upon its vertical axis, the liquid will be observed to rise from 

 the centre towards the sides to a height depending on the angular velocity and the 

 diameter of the vessel. As soon as the velocity has reached a certain limit, the liquid 

 will commence to overflow the upper edge of the vessel, being thrown from it in the 

 form of a fluid sheet in a tangential direction. If the velocity remain constant from 

 this moment, the overflow of the liquid will be observed to cease, although the liquid 

 remaining in the vessel will continue to touch the extreme edge or brim. Supposing 

 that the velocity of the vessel be now diminished, the liquid will be observed to sink, 

 but will rise again immediately to its former position^when the rotation returns to its 

 previous limit of angular velocity. This velocity is the result of the balance of two 

 forces acting on the liquid particles, namely, gravity and centrifugal force. 



It is a well-known fact that the curvilinear surface produced by a liquid in rotation is 



that of a paraboloid, the parameter of which is expressed by -^, and the curve itself 

 therefore by the formula 



f=%^^ (!•) 



X signifying vertical distance from the apex, 



y the corresponding horizontal distance from the axis of rotation, 



w the angular velocity of rotation, and 



g acceleration by gravity in one second. 



In this formula there is no factor denoting the density of the liquid, which proves 

 tiiat the point to which the liquid is raised by a given angular velocity is independent of 

 the specific gravity of the liquid employed. 



By substituting for y the radius r of the rotating cup at the brim, and for x the height 



