ACTED ON BY NO EXTERNAL FORCES. 761 



required distance from one another, the ellipsoid may be started from any position we 

 please, and the value of the divisions of the dial-plate which register the time will 

 remain invariable. 



The greater the value of X which measures the degree of divergency of the two juxta- 

 posed surfaces, the larger will be tlie divisions representing a given quantity of time ; and 

 there is no impediment to X receiving its maximum value, which is the square of the 

 least semiaxis (say c). The upper confocal surface then degenerates into a curve or 

 hoop resting upon and dri\'ing before it the rotating-plate. This gives precision to tlie 

 form to be assigned to the upper surface. Again, as regards the lower surface, whose 

 form involves two parameters, viz. the ratios of the three axes, it will hereafter appear 

 that we may without any loss of generality reduce it to depend upon a single parameter 

 by assuming the reciprocal of the square of one of its axes equal to the sum of the reci- 

 procals of the squares of the remaining two. 



Hence with a single series of ellipsoids every possible kind of motion of a free rigid 

 body may be completely represented both as regards time and place. Each ellipsoid 

 with its confocal hoop may be regarded as complete in form, the former being imagined 

 to consist of segments capable of being separated at will, so as to expose in succession 

 each part as it is wanted of the interior hoop; and by an apparatus mechanically 

 executable the motion may be followed without any break throughout the whole of 

 one or any number of periods of revolution of the instantaneous axis. 



Thus, then, the time of rotation of a free body may be kinematically determined. It 

 may also, and even more simply, be measured off by direct observation of the time which 

 a uniform ellipsoid spinning with its centre fixed upon an indefinitely rough plane occu- 

 pies in passing from one position to another. To establish this somewhat remarkable 

 law, let us consider the general case when the moments of inertia of the rolling ellipsoid 

 have any values A, B, C. The resultant of the pressure and friction which coerce the 

 ellipsoid to follow its actual path is a force always meeting the axis of instantaneous 

 rotation, and giving rise therefore to an impressed couple whose axis is perpendicular 

 to the former one. This being the case, and the ellipsoid subject to no other external 

 force, its vis viva will be constant for just the same reason as the vis viva is so in the 

 case of a system of particles connected in any manner, as by strings, whether elastic 

 or inelastic, dragging each other along one or more surfaces, and acted on by no other 

 forces except the reactions exerted by such surface or surfaces. 



To render this perfectly clear, let v^v.^v^ denote the angular velocities of the rotating 

 body about its principal axes ; X, ft, v the angles between these axes and the instan- 

 taneous axis ; J the magnitude of the couple produced by a force meeting the axis of 

 rotation, then by Euler's equations, we have 



A^— (B— C)y3U3=Jcosx, 

 ^'di~ (C— A)u,y3= J cos (i, 



