140 Prof. J, J. Sylvester on the Motion of a Rigid Body [May 17, 



half, to be pared away until it assumes the form of a semiellipsoid confocal 

 to the original surface, and that an indefinitely rough plate always remaining 

 horizontal, hut capable of turning in its own plane round a vertical axis, 

 which, if produced, would pass through the centre of the ellipsoid, is 

 placed in contact with this upper portion ; as the nucleus is made to roll 

 with the under part of its surface upon the fixed plane below, the friction 

 between the upper surface and the plate will cause the latter to rotate 

 round its axis, for the nucleus will not only roll upon the plate above, but 

 at the same time have a swinging motion round the vertical, which will be 

 communicated to the plate. I prove, by an easy application of the theory 

 of confocal ellipsoids, that the time of the free body passing from one posi- 

 tion to another will be in a constant ratio to this motion of rotation, which 

 may be measured off upon an absolutely fixed dial face immediately over 

 the rotating-plate ; and furthermore I show that the relation between the 

 angular divisions of this dial and the time depends only upon the spinning 

 force which may be supposed to set the free body originally in motion, so 

 that it will hold the same, at whatever distance, by a preliminary adjustment, 

 the rotating-plate may be supposed to be set from the fixed horizontal plane. 



Thus, then, we may realize a complete hinematical image of all the cir- 

 cumstances of the motion of a free rotating body, and reduce to a purely 

 mechanical measurement the determination of an element hitherto unrepre- 

 sented, but in reality the most important of all, viz. the tune. 



I then proceed to point out a very singular and hitherto unnoticed 

 dynamical relation between the free rotating body and the ellipsoidal top, as 

 I shall now prefer to call Poinsot's central ellipsoid, because I imagine it 

 set spinning like a top upon the invariable plane in contact with it and 

 left to roll of its own accord, the friction between it and the plane being 

 supposed adequate to prevent all sliding. I start with supposing that the 

 density of the top follows any law whatever, and call its principal moments 

 of inertia A, B, C, its semiaxes «, 6, c, the relations between these six 

 quantities being left arbitrary. 



It is easy to establish that, if a rotating body be acted on by any forces 

 which always meet the axis about which it is at any instant turning, the 

 vis mm will remain unaffected by their action ; this will be the case in the 

 present instance with the pressure and friction of the invariable plane, the 

 only forces concerned, as we may either leave gravity out of account alto- 

 gether or suppose the centre of gravity of the top to be at the centre of the 

 ellipsoid, which will come to the same thing. By aid of this principle, 

 conjoined with the two conditions to which the angular velocities of the 

 associated free body are known to be subject, it is easy to infer that the 

 velocities of this body and its representative top will, throughout the 

 motion, remain in a constant ratio, or, if we please, equal to one another, 

 provided that 



. I X I X I X 



