758 PEOFESSOE SYL\'ESTEE ON THE MOTION OF A EIGID BODY 



rigidly connected witli the body, and which may be termed its kinematical exponent, is 

 supposed to have its centre fixed, and to turn with a purely rolling motion upon a plane 

 in contact with it which contains the constant impulsive couple L, capable at each 

 moment of time in any position into which the body has turned, of communicating to it 

 from rest the motion Avhich it then actually possesses. If we suppose that the angular 

 velocity of rotation is always equal to LPE., where P is the length of the perpendicular 

 distance of the fixed centre from the tangent plane, and R is the length of the radius 

 vector drawn from it to the point of contact, the path and velocity of the motion of the 

 body in rigid connexion with the ellipsoid is completely represented ; this is Poia'sot's 

 theorem stated in its complete form. 



To fix the ideas, let us consider the iuAariable plane to be horizontal ; if we were to 

 apply a second plane parallel to the former fixed one, and also touching the ellipsoid, 

 this would in no respect aflfect the motion — the ellipsoid might be made to roll between 

 the two planes instead of rolling upon the under one alone ; but if we were arbitrarily 

 to alter the form of the upper part of the surface, the motion of rolling would in general 

 be no longer possible ; the only motion that could take place would be that of swinging 

 ■round the vertical axis perpendicular to the two planes. In order that the ellipsoid may 

 be able to roll as well as to swing, a certain geometrical condition must be satisfied, viz., 

 the plane passing through the radius vector from the centre O to R, the point of contact 

 with the given plane, and through the vertical perpendicular in question PO/), must 

 contain the point of contact r of the upper surface with the upper plane ; for then, and 

 then only, the rotation about OR may be resolved into two rotations about Or, Op 

 respectively, and the ellipsoid whilst it rolls about OR, will be swinging round O}) 

 [or it may obviously at the same time be rolling and swinging (the latter in unequal 

 degrees) upon each of the parallel tangent planes] ; if this condition were not fulfilled, 

 •the ellipsoid, in the act of rolling upon the lower plane according to the direction of its 

 motion, would either quit the upper one or tend to force it upwards ; but as the upper, 

 like the lower plane is supposed to be at a fixed distance from the centre, this tendency 

 would be resisted, and thus the supposed motion of rolling upon the lower plane without 

 quitting contact with the upper one could not be realized. 



The condition that OR, POp, Or shall lie on one plane, we have seen will be fulfilled 

 if the upper surface be a portion of an ellipsoid confocal with the lower one, and in 

 that case the body may remain continually in contact with both planes whilst it rolls 

 on the lower one ; and we have thus a complete solution of the kinematical problem of 

 determining what form must be given to the upper part of a body, the lower portion of 

 whose surface is ellipsoidal, in order that it may be able to roll as well as swing between, 

 and in contact with, two parallel fixed planes. 



Call, then, the squared semiaxes of the lower surface a", b^, <?, and those of the upper 

 one a^— X, 5"— X, c'— X, and let us proceed to calculate the respective values of the two 

 rotations about Op, Or equivalent to the single rotation LPR about OR. 



In PO, RO produced set off" OP,, OR, equal to OP, OR, and draw 'Rii\ parallel to 



