580 
ROTATION FOR STABILITY OF 
Since T does not contain x } y , z, therefore 
s l = 0 M = o ^ = 0- 
Sx ’ St, ’ Si ’ 
and therefore equations (1), (2), (3) become 
(S)-».|(D-»4(S)-»' 
d /ST' 
dt 
The integrals of these equations may he written 
ST _ ST ST _ 
U- 0 ’Sj,=°'Wz- F ’ " 
(7) 
a constant, the total momentum of the system, by taking the axis of z 
in the direction of the resultant momentum. 
Now, the components of linear momentum of the system in the 
directions OA, OB, OC are c n u, c n v, c S3 w, and therefore 
CiiU = F cos AZ == — F sin 0 cos <£, .(8) 
c^v = F cos BZ = F sin 6 sin <£, ..(9) 
c^w = F cos CZ = F cos 0 ......(10) 
Now, the component velocities x, y, z are equivalent to the component 
velocities k cos if/ + y sin if/, — x sin if/ 4- y cos if/, z in the directions 
OF, OF, OZ; which, again, are equivalent to the component velocities 
(x cos \J/ + y sin xj/) cos 0 — z sin 6 in the direction 01), 
— x sin if/ + y cos if/ ,, « OF, 
(x cos \f/ + ij sin if/) sin 6 + b cos 0 ,, „ OC ; 
which, again, are equivalent to the components 
{{x cos if/ + y sin \J/) cos 0 — z sin 6} cos cf> + (— x sin if/ + \j cos if/) sin <f> 
— (cos 0 cos <£ cos if/ — sin sin if/) x + (cos 0 cos <£ sin if/ + sin d> cos ifr) y 
— sin 6 cos 4>z 
= u, in the direction OA; 
— {(a? cos \J/ + y sin if/) cos 6 — z sin 6} sin cf> + (— x sin if/ + y cos if/) cos $ 
= — (cos 6 sin cos if/ + cos sin if/) x — (cos 6 sin <£ sin if/ — cos cf> cos if/) y 
4- sin 0 sin <f>z 
= v, in the direction OB ; and 
(x cos if/ + y sin if/) sin 0 + b cos 6 
= sin $ cos if/x 4- sin 6 sin if/y + cos Ob 
= w , in the direction OC. 
