760 PROFESSOR SYLVESTER ON THE MOTION OF A BIGID BODY 



i. e. the difference between the squared velocities of any two bodies of the set is con- 

 stant throughout the motion. 



The above is a theory of rigid bodies whose kinematical exponents are confocal ellip- 

 soids, and it has been shown that the motion of the Avhole set of bodies thus related, 

 both as regards position and velocity, is completely determined when we know tlie 

 motion of any one of them. It will hereafter appear from the analytical treatment 

 of the subject that an analogous theorem applies to bodies whose kinematical expo- 

 nents, instead of being confocal, are what may be termed contrafocal ellipsoids ; ellip- 

 soids, that is to say, the sums instead of the differences of whose squared axes aie the 

 same in all three directions. 



By turning an ellipse through 90" round its centre we obtain a contrafocal ellipse ; 

 and contrafocal ellipsoids will be those all of whose principal sections are contrafocal. 



To every infinite series of confocal ellipsoids there will correspond another such 

 series, each ellipsoid of one series being contrafocal to each of the other, and it may 

 very easily be seen that no two ellipsoids taken respectively out of the two opposite 

 series can be obtained from each other by a mere change of place, as is the case with 

 contrafocal ellipses ; so in the instance of binary covariants and contravariants, any such 

 can be converted into each other by the simple interchange of x, y with y, — :r, but no 

 such or similar commutability exists between covariants and contravariants of the 

 ternary species. It may be here convenient to notice that the kinematical exponent 

 (or momental ellipsoid) of a given uniform ellipsoid is not the ellipsoid itself, but the 

 reciprocal of the contrafocal ellipsoid whose squared semiaxes are X—a^, X — b^, a — tf, 

 where X=a''-\-b''-\-c''. 



It is now clear how the time of passage from one position to another is susceptible of 

 mechanical measurement. Let the upper part of Poinsot's ellipsoid, whose semiaxes are 

 a, b, c, be pared away until it assumes the form of a segment of an ellipsoid whose 

 squared semiaxes are a^— x, b^—X, c-—X; let the linear sui'face be in contact with a 

 rough plane absolutely fixed, whilst its upper surface is so with a parallel jjlafe not 

 absolutely fixed, but capable of turning round an axis perpendicular to the two planes, 

 and which if produced would pass through the centre of the ellipsoid. Then, when by 

 the hand or any mechanical contrivance the body is made to spin like a sort of top upon 

 the lower plane, it will also spin upon the plate above, and at the same time by the 

 friction drive it round the vertical axis ; the angle of rotation round this axis will give 

 the exact measure of the time which the free body ideally associated with the ellipsoid 

 would occupy in passing from one position to another. If this angle (which of course 

 may be made to register itself by the motion of a hand upon a fixed dial-plate immedi- 

 ately over the rotating one which carries the index) be called <p, the time in question 



will be j-^, where it is particularly deserving of notice that the denominator L>. is 



independent of the initial position of the body ; hence by supposing the plane and 

 rotating-plate to be capable by a preliminary adjustment of being shifted to any 



