9, 
THE KEY. N. M. FERRERS ON PROFESSOR SYLVESTER’S 
jection of the instantaneous axis on the rough plane is denoted by [a, and the whole 
angular velocity by co, so that we have 
yj + u 2 — <■ 
AYe have also the following relations : — 
ur. up ur A 2 
I_|_ 2 _(__!— _ 
a 2 b~ c 2 
r 
4. t w 3 _^ 
Z.4*l „4 .. 4 * 
b 4 c 4 p 
And the principal moments of inertia are represented by 
( 2 ) 
( 3 ) 
( 4 *) 
/G H\ /G H\ 
ya 2 p 2 j C ’ y b 2 p 2 J 
„2 m a)5 2 C", ( /i2 ^2 ) C 2 « 2 , p^ja~b~, 
which will be found, when substituted for A, B, C, to satisfy the relation already stated. 
In the case of a uniform ellipsoid, we have 
/I 1 JA H 
{ *\a 2 +b 2 +c 2 )-J 2 ' 
* [The equations (1), (2), (3), (4), and the invariability of A, may also he proved as follows. If x, y, z be 
the coordinates of the point of contact, referred to the principal axes, we have 
Also 
whence 
K+£+*= 1 , A&’+YL 
a" b~ c a b c p 
x y z _p 
«h W 2 “>3 x ’ 
w\ ,u]\ . u)\ A 2 eof ew| o>3 A 2 . 
a~ lr c 2 p~ ct l b l c l ~~ p l 
H 
Multiply these equations by — — , G, and add, then 
a 2 bP 
A 2 _ p' vis viva of the ellipsoid 
G=H ’ 
and is therefore constant. 
Hence the direction-cosines of the perpendicular to the fixed plane are 
^\P_ vpl 
a~ a’ b 2 . 
and since this line is fixed, 
1 p 2 cho l 
a~ A dt 
or 
1 f/Wj / 
Similarly 
or dt V< 
1 du> 2 f 
b 2 dt 1; 
1 dio 3 / 
c 2 dt \£ 
A W3 c 2 A 
?)' A=0 - 
0. — February 1870.] 
