REPRESENTATION OF TIIE MOTION OF A FREE RIGID BODY. 
[We shall first investigate the value of h K , the component angular momentum of the 
ellipsoid about the normal to the rough plane. The cosines of the inclinations of this 
7)^ CO T)~ 00 T)^ OO 
line to the principal axes of the ellipsoid are ^ y- J respectively ; and the 
A a 2 ’ a b 2 ’ A c“ 
component angular momenta about the principal axes are 
/G H\^ 2 /G H\ 2 2 /G H\ „, 2 
\a*~~ f) h C \b 2 f) cac °v \c* 
respectively. Hence 
7 p*u G H\5 9 c* 2 , /G H\cV 2 , /G H\« 2 6 2 
1k A | ya- p 2 J a 2 ylj 2 p 2 ) b 2 yc 2 p 2 J c 2 i 
Now, by multiplying (2), (3), (4) by - [A+A+A)’ (1+P+?) res l’ ec ' 
tively, and adding, we see that 
2 2 
COli&2 l_*"3 
-s-rurr--fi 
b 6 
Hence 
AL_A 2 , 2 / 1 _i\ ( l _ i\ /I_I\ i 
c 6 / yp 2 a 2 )\p 2 b 2 )\p 2 c 2 / _r « 2 6 2 ( 
(H(? 
a 2 b 2 c 2 
A' 
2 
1 )(i?- 1 )K G ‘"' 2 - H /“ w 
] 
= (G - H)X + G*p* (l (l (l -f) +G ^ (5) 
We shall next investigate a relation between the component angular momentum of 
the ellipsoid about any axis through its centre, and that, about the same axis, of a 
particle of mass G, situated at the point of contact of the ellipsoid and rough plane, 
and moving as that point moves. If /, m, n be the direction-cosines of the axis referred 
to the principal axes, the component angular momentum of the body about it is 
or 
(? 2 “ ; 7 2 ) (p— p) cVm» a + Js) a-b-nu. 
(I ® i , mco a , ncti 3 \ H //oj, mai 2 mo 3 \ | 
\« 4 + b 4 + (AJ ~~p 2 ^fl 2 b 2 "r c 2 
cftfcfl G 
( 6 ) 
Again, the coordinates of the point of contact are y <y 15 y t» 2 , o 3 respectively. Hence 
its component velocities, parallel to the axes, are 
p dw j p dco 2 p dco 3 m 
A If 1 A ~dt ’ A ~ctt’ 
and its component angular momenta are therefore 
dco, 
r f ( J“>3 
U7o S',, -57 - 
A 2 \ 2 dt 
■co. 
dt 
\ n P*( (Im ) da >z\ n P* ( 
)’ G A 2 y 3 It ~ a <Tt)' G 7 2 ^ 
Aoj 
2 dt 
B 2 
