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XXV. On the Motion of a Bigid Body acted on hy no external Forces. 

 By J. J. Sylvester, LL.B., F.B.S. 



Received April 26,— Ecad May 17, 1866. 



As conveying an image of the motion of a rigid body acted on by no forces, Poiksot's 

 well-known method of representation, whether by a rolling ellipsoid or a shifting cone, 

 labours under an obvious imperfection ; the time is not put in evidence by it. Thus 

 when the ellipsoid, with which alone I intend here to deal, is employed, it is true that 

 the proportional value of the velocity of rotation about the instantaneous axis is geome- 

 trically measured by the radius vector drawn from the fixed point to the invariable tangent 

 plane, and so by a process of summation the time of passing from one position to another 

 may be considered as inferentially determined ; but there is nothing to convey to the 

 senses, or to the mind's eye, a notion of the effect of this summation, and thus the rela- 

 tion of the most important element — the time — to the position of a free revolving body 

 remains unexpressed. I shall begin with showing how by a slight addition to Poixsot's 

 ideal kinematical apparatus this defect may be completely removed, and the time 

 between successive positions conceived to register itself mechanically. As the projjerty 

 upon which this depends readily lends itself to a geometrical form of proof, I shall, in 

 the first instance, follow that mode of investigation, as being the more germane to the 

 matter in hand, reserving to a later point in the memoir the analytical demonstration ; 

 that is to say, assuming Poinsot's ellipsoid, and the law which connects the velocity 

 with the position of the body, I shall show how the time may be, as it were, mecha- 

 nically extracted and summed. 



It will be well, then, in the first instance to recall some simple properties of confocal 

 ellipsoids which I shall have occasion to employ. If parallel tangent planes be di-awn 

 to a system of confocal ellipsoids, it is well known (see Dr. Salmon's great work on 

 Surfaces, Art. 202, 1st edition, or Art. 184, 2nd edition) that the points of contact lie in 

 a plane curve, and that this curve is an equilateral hyperbola. Since a concentric sphere 

 with an infinite radius belongs to the system of confocal ellipsoids supposed, it follows 

 that the point of intersection of the perpendicular from the centre of the ellipsoid 

 upon the tangent planes with the plane at infinity, is a point in this curve, or, in other 

 words, such perpendicular is contained in the plane of the hyperbola, and is an asymp- 

 tote to the latter. The above is all that is required to establish the dynamical theorems 

 necessary for my immediate purpose. 



The revolving body being assumed to have moments of inertia A, B, C about the 

 principal axes, the ellipsoid 



Aa'+By-+Cs==l 



MDCCCLXVI. , 6 h 



