120 
ROTATION FOR STABILITY OF 
Describe a sphere of unit radius with centre 0, and let OZ be the 
direction of the resultant linear momentum Z, OG of the resultant 
z 
angular momentum 0 of the system, OC of the axis of figure. (In the 
figure the eye is supposed to be at 0, and looking at the concave side 
of the sphere—just as the eye sees the concave side of the celestial 
sphere.) The angular velocities and r are estimated on the right- 
handed screw system (that is, an angular velocity r about OC is 
reckoned positive when on a right-handed screw it would cause a 
transference from 0 to C). 
If the centre, 0, of the body had been fixed, then 00, the axis of 
resultant angular momentum, would have been fixed, and the body 
would have behaved as if the equatoreal and polar moments of inertia 
were c 4 and c 6 ; the axis OC would have described a right circular 
cone about 00 as axis; and the motion might have been represented 
by rolling the right circular cone of axis OC and semi-vertical angle 
IOC, fixed in the body, on the right circular cone of axis 00 and 
semi-vertical angle 100 , fixed in space; 01 being the instantaneous 
axis of rotation, and therefore 
tan IOC = - e tan GOO. 
c i 
But when the body moves steadily in the medium under no forces, 0 
describes a uniform helix about a fixed straight line parallel to OZ, 
while 00, 01, and OC lie in a plane passing through OZ, which 
revolves with uniform angular velocity (/x, suppose), while OC makes a 
constant angle (a suppose) with OZ, and 00 makes a constant angle 
(6 suppose) with 00* 
