PHILOSOPHICAL TRANSACTIONS. 
I. Note on Professor Sylvester’s representation of the Motion of a free rigid Body by 
that of a material Ellipsoid whose centre is fixed, and which rolls on a rough Plane. 
By the Bev. N. M. Ferrers, Fellow and Tutor of Gonville and Caius College , Cam- 
bridge. Communicated by Professor J. J. Sylvester, F.B.S. 
Received May 29, — Read June 17, 1869. 
Ix a paper published in the Transactions of the Royal Society for 18G6, Professor Syl- 
vester has given an important extension of Poinsot’s representation of the motion of a 
freely rotating rigid body, by means of the momental ellipsoid. lie has proved that if a 
material ellipsoid, similar in form to the momental ellipsoid, and so constituted that its 
principal moments of inertia, A, B, C, are connected with its semiaxes a, b , c by the rela- 
tion A a i {lr — P) -\-¥>b\c~ — a 2 ) + CV(« 2 — & 2 ) = 0, be made to roll in contact with a perfectly 
rough plane, the motion of this material ellipsoid will be precisely the same as that of 
the momental ellipsoid of the rigid body ; the rough plane taking the place of the geo- 
metrical fixed plane, in contact with which the momental ellipsoid is supposed to roll. 
He has also investigated expressions for the pressure and friction between the ellipsoid 
and the rough plane, in terms of the angular velocity of the ellipsoid, and of the length 
of its axes, and the distance of the centre from the rough plane. In investigating inde- 
pendently the values of these forces, I have been led to a somewhat different treatment 
of the same problem, in the course of which some theorems have presented themselves 
which may be not without interest. 
The notation which I adopt is as follows : <y 2 , co 3 represent the component angular 
velocities of the ellipsoid about its principal axes, which, as proved by Professor Syl- 
vester, are connected by the following equations : — 
The distance from the centre of the ellipsoid to the rough plane is denoted by p , and 
the component angular velocity of the ellipsoid about the normal to the rough plane, 
which is known to be constant, by a. The component angular velocity about the pro- 
mdccclxx. E 
