6 
THE KEY. N. M. FERRERS OX PROFESSOR SYLVESTER’S 
and 
Again, 
ittJ? — y/jjj -j- aj. 3y — j 5 
l Jj ' Qlfi~ n2 ) = ~ V - i 1 +^ r / + + “fifty — 2a/3yX 4 — u*(3>Y p, 
P=Gpj — 2f/, 2 — (l+fiy+yci + ccfi)}, 2 — 2ufiy^— cffiY pj. . . . 
F=Gp|, 
( 13 ) 
~ 7^] “ (^ 2 +/ 3r A 2 )(^ 2 + + «/3 a 2 )| • ..... (14) 
[The theorem contained in equations (5) and (6) may perhaps receive additional illus- 
tration by a comparison of the moments, about the principal axes, of the forces acting- 
on the ellipsoid, and of those acting- on the particle coinciding with the point of contact. 
Since the component angular momenta of the ellipsoid about the principal axes are 
it follows that the moment of the forces 
about one of the principal axes is 
Ire 
2 dwi j /G H 
dt 
X 
b 2 
p 
or 
b q c~ da )j 
« 2 dt 
( tree (rb 2 \ ) 
yw—f J" 2 " 3 } 
h 
a result independent of H. Now, if we refer to equation (7) we shall see that the 
angular momenta of the particle only differ from those of the ellipsoid by having G 
written in place of H ; consequently the moments of the forces, since they do not involve 
IT, must be the same for the particle and the ellipsoid. It follows of course that the 
moments of the forces about any other axis must be the same in both cases. 
In the above investigation of the value of P, I have followed Professor Sylvester, in 
assuming that the friction acts wholly in a direction perpendicular to the instantaneous 
axis, The other component of the friction is necessarily indeterminate, since any force 
in the direction of the instantaneous axis may be combined with it, without altering its 
effect. I have assumed this component of the friction to be zero ; if it be taken to be 
equal to an arbitrary force F', the value of P above investigated must be increased by 
— . The values of the moments of the forces are not, of course, affected by this suppo- 
sition ; and if F' be so chosen that the pressure between the ellipsoid and the rough 
