80 Trans. Acad. Sci. of St. Louis. 



It should, again, be observed that here, as in Case //, the 

 condition ^'o = is compatible only with the condition w < 1. 



Remark. The motion of the axis N^' of the torus includes 

 the motion of the ring {G^). In the cases I A, IC, II A, 

 III Ay this ring oscillates about PP' ; in the cases IB and 

 IIB it assymptotically approaches the ring (Cj); finally, in 

 the case IIIB, it revolves about PP ' without a change in the 

 direction of this revolution. 



To complete the determination of the motion of the gyros- 

 cope it is necessary to evaluate the angles ^ and t/t. The 

 first angle defines the motion of the ring ( CJ about ZZ', the 

 second determines the motion of the torus about its own axis 

 NN '. These angles are given by the formulas 







(23) <^ = <^„_a,^+ I \~:^\''^,l^ dt 



d-\- b sm^ 6 



(24) 1/r 



i d -\-b sm" 



In the following discussion the terms precession and nuta- 

 tion will be used to indicate the rates of change of the angles 

 4> and 6 respectively. The problem which we set ourselves 

 to solve is to determine when the motion of the gyroscope 

 proceeds so that the nutation is = 0. 



The condition necessary and suflGlcient to obtain a motion 

 without nutation is that 



sin 0^ (?2 — CI, cos 0^) \_{h-\-d) CI, — bl, cos ^o] = 

 or, on account of the relations (16), 



(25) sin 0, {CO + (f>',) [CI,- b COS 0,((o + <!>',)•]= 0. 



This equation may be satisfied in three ways. 



