89] 



APPLICATIONS OF SYMMETRICAL TOP. 



273 



constitutes the directrix of the herpolhode cone. Since the ellipsoid 



of inertia is of rotation, the axis of figure OF, 



the instantaneous axis 01, and the axis of 



angular momentum OH, lie in the same plane, 



which is perpendicular to the tangent plane to 



the herpolhode cone. During the rolling, all 



these axes move parallel to this tangent plane, 



so that the vector HH L , representing the change 



of angular momentum, is parallel to the tangent 



plane, and in the direction of advance of the 



axis of figure. The couple causing the motion 



accordingly due to the reaction between the 



wire herpolhode and the top, is always parallel 



to the tangent plane, and never vanishes, but 



always tends to press the top against the wire. 



Or in general, in constrained motion, the motion 



causes the polhode cone to press against the herpolhode cone. This 



seems to have been first explicitly stated by Klein and Sommerfeld, 



Theorie des Kreisels, p. 173. 



An application of the above 

 principle on a large scale, and the 

 only one known to the author, is 

 found in the Griffin grinding mill. A 

 massive steel disk or roller A (Fig. 89) 

 hangs from a vertical shaft by a uni- 

 versal or Hooke's joint C, in the middle 

 of a steel ring B forming the side of 

 a pan. If now the shaft be set rotat- 

 ing, the roller spins quietly about a 

 fixed axis, with no tendency to move 

 sidewise. If on the contrary it be 

 brought into contact with the ring, 

 it immediately rolls around with great 

 velocity, pressing with great force 

 against the steel ring or herpolhode, 

 and grinding any material placed in 

 the pan with great efficiency. It is 

 interesting to note that a somewhat 

 similar mill, in which the axis, instead 

 of passing through a fixed point, hangs 

 vertically from a revolving arm, and 

 therefore is devoid of the action just 



described, although both mills possess in common the centrifugal 

 force due to the circular motion of the center of mass of the roller, 



WEBSTER , Dynamics. 18 



Fig. 89. 



