342 



III. The equation of relative moments round the polar line, 



2 (m.^ 2 f ^t) io). (3) 



dt J V Jo 



Where r = projection of radius vector from the origin to any element on 

 a plane perpendicular to the polar line, 



= angular velocity of this projection. 



do 



This equation can be very easily proved from the consideration of Corioli's 

 forces ; but it is unnecessary to resort to them, for it is evidently but 

 another form of the equation of the conservation of absolute moments 

 round the same line, 



dt I V Jo 

 since 



absolute ^ = relative + 

 at dt 



'Now, let G = moment of inertia round axis of figure, 



A = same round any axis perpendicular to this, 

 C 



— n = m; 



then, since the relative motion of the gyroscope may always be resolved 

 into two, its [apparent] rotation round its own axis, n - lo cos 0, and an 



ds 



angular velocity — round an axis at right angles to its own axis, 

 do 



the relative ms viva =Ai~j + C{n- iv cos oy. 



Also /= C cos W + A sin ^9 = (C-A) cos '0 + A; 

 .*. equation (2) assumes the form 



A C{n-io cos ey = io^ {€- A) cos W + Const. 



Or, 



— j = 2 cos 6> - w2 cos + Const. (4) 



If the axis is restricted so as to be compelled to trace out a particular 

 curve on the unit sphere, the equation of this curve gives another rela- 

 tion between (s) and {0), which combined with this determines the 

 motion. 



