of a rigid Body round djixed Point, 433 



Some years ago, however, this defect of the theory was re- 

 moved in a memoir presented to the French Institute, by an 

 author remarkable as well for the originality as the perspi- 

 cuity of his views, in which he reduces the motion of the body 

 to that of an ellipsoid whose centre is fixed, rolling on a fixed 

 plane. 



In the very brief extract which has been published of this 

 memoir, Poinsot assumes an ellipsoid whose centre coincides 

 with the fixed point, and whose semiaxes are proportional to 

 the inverse square roots of the moments of inertia, round the 

 three principal axes passing through the fixed point, coin- 

 ciding with the axes of the ellipsoid ; and thence by the 

 known geometrical properties of this surface, and the tangent 

 plane, deduces the nature of the motion, with great elegance 

 and simplicity. 



In the following pages an ellipsoid is assumed different 

 from that of Poinsot, in which the squares of the axes are 

 proportional to the moments of inertia round the principal 

 axes of the body passing through the fixed point, coincident 

 with the axes of this ellipsoid, which may be termed the 

 Ellipsoid of moments. 



It becomes proper to mention, that as the two ellipsoids of 

 moments, that assumed by Poinsot, and the one here adopted, 

 are reciprocal surfaces, all the properties of such motion may 

 be indifferently deduced on either system, and then at once 

 transferred to the other, by the known properties of recipro- 

 cal surfaces ; a few examples of such transformation are given 

 towards the close of this paper. 



It appears, however, the more eligible course to deduce 

 the general properties of rotatory motion, directly and inde- 

 pendently, from the fundamental and acknowledged principles 

 of mechanics, than to have recourse to the aid of reciprocal 

 surfaces, with the relations of which many readers may not 

 be familiar. 



Previous to entering on the subject to which this paper is 

 more particularly devoted, it may be well to state and prove 

 a few general propositions bearing on the subject. 



II. Let Ox, Oy, O z, be three rectangular axes passing 

 through the fixed point O, the axe O * being the instan- 

 taneous axis of rotation of the body*, X, Y, Z the forces which 

 act on any particle d m of the body, of which the coordinates 

 are x y z ; these forces being translated to the origin, are there 

 equilibrated by the resistance of the fixed point O, while 



* The existence of an axis of instantaneous rotation is assumed, as a 

 proof of it from elementary principles may be easily given. (See Earn- 

 shaw's Statics, art. 109.) 



Phil. Mag. S. 3. Vol. 19. No. 126. Dec. 184-1. 2 F 



