345 



Thikd Case.—TA^ Axis is unrestricted. 



Denoting as before the polar line and the axis of the gyroscope by 

 P and X, let the angle which the arc (PX) makes with a fixed arc 

 through (P) = ; the relative angular motion of the gyroscope may be 

 resolved into three rotations : — 



1 



o) cos 6 round X ; 



de 



r n 



I sin 0 round an axis in plane PX at right angles to (X) ; 



U/t 



round an axis perpendicular to plane ( OP). 



dt 



IS'ow, by the equation (3) of relative moments round ( 0), 

 sin ^ . A^m 0~ + Go^ 0 . C {n - oo co^ 6) ^- {C - A) tv cos "^9 = Const.; 



U/t 



or, if the axis be started at relative rest, 



Sin 261 ^ = - m (cos e - cos 6>o) + w (cos ^0 - cos Wq, (7) 



and by the equation (4) of relative vis viva^ 



Sin'6>^^"^ + = 2mw (cos 6 - cos 6>o) 



- (xfi (cos ^0 - cos (8) 



multiplying (7) by (2a;), adding it to (8), and writing Y^'for + oot, we 

 obtain 



On making the same substitution in (7), it becomes 



Sin ^0 ~- = m (cos 6>o - cos 6) + w sin ^Oq. (10) 



(Y^O evidently represents the angle the arc (PX) makes with an arc 

 through P retreating with an angular velocity (uj) ; and the equations 

 (9) and (10) between (0) (Y^') and {t), are those of the curve described 

 by the axis of the gyroscope with respect to this retreating co-ordinate 



