3T0 



centre, and OA, OB, OC for the directions of their semiaxes. 

 And the attraction of such a shell on an external point may be 

 simply expressed by means of the semiaxes of a confocal ellip- 

 soid passing through the point. (See a Memoir by M. Chasles 

 in Liouville's Journal, vol. v.) The quantities which we have 

 called p and p' are, in fact, semiaxes of an ellipsoid described 

 through the attracted point (that is, through C in the first 

 case, and through B in the second) so as to be confocal to the 

 surface of which the semiaxes are acos^, bcoscj), ccos^." 



A note by Professor Mac Cullagh, on the rotation of a 

 solid body, was read. 



Let a solid body be made to revolve round a fixed point 

 O, and be afterwards free from any external forces; and 

 through O conceive a right line 01 to be drawn perpendicu- 

 lar to the invariable plane (the plane passing through O and 

 the direction of the primitive impulse). If O be the centre 

 of an ellipsoid whose semiaxes are in the directions of the 

 principal axes belonging to that point, and of such lengths 

 that the square of each semiaxis is equal to the corresponding 

 moment of inertia divided by the mass of the body, the motion 

 will take place in such a way that the point I, in which the 

 right line 01 intersects the surface of the ellipsoid, will be 

 fixed in space ; and therefore O I will describe within the 

 body a cone of the second order, condirective with the ellip- 

 soid (that is, having its circular sections parallel to those of 

 the ellipsoid), while the point I describes on the surface of the 

 ellipsoid a certain spherical conic. In a former number of the 

 Proceedings (vol. ii. p. 542), the author had alluded to a theo- 

 rem for determining the time at which the point I occupies 

 any given position on the spherical conic, and he now gave a 

 particular statement of it as follows : 



Conceiving a plane of circular section of the ellipsoid to 

 be drawn through its mean axis, and the spherical conic to be 

 projected on this plane, first by right lines parallel to the 



