370 
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@, bcos, 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 OI 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 OI intersects the surface of the ellipsoid, will be 
fixed in space; and therefore OI 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 
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