1866.] On the Motion of a Rigid Body^ ^c. 139 



Section YIII. Certain apparent exceptions are considered, in which it 

 is not found possible to express the symbols of chemical substances by- 

 means of an integral number of prime factors, consistently with the as- 

 sumption of the modulus a. 



The Society then adjourned to Thursday, May 1 7. 



May 17, 1866. 

 Lieut. -General SABINE, President, in the Chair. 

 The following communications were read : — 



I. "On the Motion of a Rigid Body moving freely about a Fixed 

 Point.'^ By J. J. Sylvester, LL.D., F.R.S. Received April 

 1866. 



The nature of the present brief memoir will be best conveyed by my 

 giving a succinct account of the principal results which it embodies, in the 

 order in which they occur. The direct solution, in its present form, of the 

 important problem of the motion of a rigid body acted on by no external 

 forces, originating in the admirable labours of Euler, has received the last 

 degree of finish and completeness of which it is susceptible from the powerful 

 analysis of Jacobi ; in one sense, therefore, it may be said that the discussion 

 is closed and the question at an end. Notwithstanding this, in the mode of 

 conceiving and representing the general character of the motion, there are 

 certain circumstances which merit attention, and which may be expressed 

 without reference to the formulae in which the analytical solution is con- 

 tained. 



Poinsot's method of representing the motion by means of his so-called 

 "central ellipsoid" has passed into the every-day language of geometers, 

 and may be assumed to be familiar to all. The centre of this ellipsoid is 

 supposed to be stationary at the point round which any given solid body 

 is turning ; its form is determined when the principal moments of inertia 

 of that body are given, and it is supposed accurately to roll without sliding 

 on a fixed plane whose position depends on the initial circumstances of the 

 motion. The associated free body is conceived as being carried along by 

 the ellipsoid, so that its path in space, its continuous succession of changes 

 of position, is thereby completely represented ; but no image is thus pre- 

 sented to the mind of the time in which the change of position is effected. 

 1 show how this defect in the representation may be remedied, and the time, 

 like the law of displacement, reduced to observation by a shght modification 

 of the apparatus of the central ellipsoid or representative nucleus, as it will 

 for the moment be more convenient to call it. To steady the ideas, imagine 

 the fixed invariable plane of contact with the nucleus to be horizontal and 

 situated under it ; now conceive a portion of its upper surface, say the upper 



