1869.] 



On the Motion of a free rigid Body. 



471 



June 17, 1869. 



Lieut. -General SABINE, President, in the Chair. 



Mr. J. Ball, Mr. J. N. Lockyer, and Vice-Admiral Sir Spencer Robinson 

 were admitted into the Society. 



The following communications were read : — 



I. "Note on Professor Sylvester's representation of the Motion of 

 a free rigid Body by that of a material Ellipsoid rolling on a 

 rough Plane." By the Rev. N. M. Ferrers, Fellow and Tutor 

 of Caius College, Cambridge. Communicated by Professor J. J. 

 Sylvester. Received May 29, 1869. 



(Abstract.) 



This paper is intended as a sequel to Professor Sylvester's paper above 

 mentioned, which was published in the Philosophical Transactions for 

 1866. The notation, so far it differs from Professor Sylvester's, is as 

 follows : — 



p is the distance from the centre of the ellipsoid to the rough plane. 



\ the (constant) component angular velocity of the ellipsoid about the 

 diameter normal to the rough plane, fi the component angular velocity of 

 the ellipsoid about the diameter parallel to the projection of the instan- 

 taneous axis on the rough plane. 



A p *are the component angular momenta about these diameters respec- 

 tively. 



h t about the diameter at right angles to both. 



n the angular velocity, in space, of the plane through the instantaneous 

 axis perpendicular to the rough plane . 



Then the mass of the ellipsoid being taken, as in Professor Sylvester's 

 paper, to be unity, it is proved that 



A 



The following theorem is then established : — " The component angular 

 momentum of the ellipsoid about any diameter parallel to the rough plane 

 is equal to^, multiplied into the component velocity of the point of contact 

 of the ellipsoid and rough plane, in the direction at right angles to this 

 diameter." 



It hence follows that 



-hence it is proved that 



VOL. XVII. 2 N 



dt p 



