Chessin — On the Motion of Gyroscopes. 29 



A. If n < 1 The axis of the torus oscillates about ZZ'. 

 These oscillations are isocJironic and their amplitude is 2rj^ 



where rj^ = cos"^ n. The period of a complete oscillation is ^r^ 

 and 



/^2_ 

 ]/( 

 COS 

 



, - , V d + b sin' e dd 



(20) t3 = x|?aI ^ 



2 y' n — cos 6 



B. If n ^ 1, the axis of the torus assymptotically approaches 

 the position OZ' (where 6 = tt). 

 Case III. m < — \ 



Let us put — m = 5, so that s > 1. Then 



l/d-\-hBm^e dd 

 dt ' 



= ±VI 



1/(5 + cos 6) {n — cos^) 



A. If n < 1, the axis of the torus performs isochronic oscil- 

 lations about ZZ'. The amplitude of these oscillations is ^t]^ 

 where rj^ = cos~^ n, and the period of a complete oscillation is 

 4X4, having put 



/V 



.21) I" I Vd + bsin'^ede 



|/ ( s + cos 6) (n — cos 6) 



B. If n > 1, the axis of the torus revolves about PP' with- 

 out changing the direction of the revolution. The time re- 

 quired for a complete revolution is 2r^ where 



TT 



M 



(22) r,= ^r-.' v/rf + isinM de 



\/{s + cos 6) (71 — COS 6) 



