Chessin — On the Motion of Gyroscopes. 2 9 



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

 These oscillations are isochronic and their amplitude is 27;, 



where Vi = co8~^ n. The period of a complete oscillation is ^t^ 

 and 



/V2 



i/d + b sin" e d0 



(20) ^. = ^(2A I » 



COS - -J/ yj — cos 



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

 the position OZ' (where = it). 

 Case III. m < — 1 



Let us put — m = 5, 80 that 5 > 1. Then 



l/d-\-hs\n^0 d0 

 dt 



= ±>|l 



1/(5 + cos 0) {n — cos 0) 



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

 lations about ZZ' . The amplitude of these oscillations is 2r}^ 

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

 4r^y having put 



^.=vi/; 



^21) - . i/d-¥bsm'0d0 



]/ ( « + cos 0) {n — cos 0) 



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 2t^ where 



-^If 



_ _ ," . i/d-\-b6m^0 d0 



l/(« + cos 0) {n — cos 0) 



