30 Trans. Acad. Set. of St. Louis. 



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

 condition 0'^ = O'ls compatible only with the condition n < 1. 



Remark. The motion of the axis iVTV' of the torus includes 

 the motion of the ring ((7J. In the cases lA, IC, IIA, 

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

 IIB it assymptotically approaches the ring (C^); 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 y^r. The 

 first angle defines the motion of the ring ( C^) about ZZ', the 

 second determines the motion of the torus about its own axis 

 NN '. These angles are given by the formulas 



S' 



l^ _ CI, cos d 



(23) * = *,-..+ I ';,^,^„., ^^ 



^.-. , .17. i (lo — CZ, cos ^) cos ^ ,, 



In the following discussion the terms precession and nuta- 

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

 (f) 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 sufficient to obtain a motion 

 without nutation is that 



sin 0,(1^— CI, cos e,) [(6 + d) CI, — bl, cos ^o] = 



or, on account of the relations (16), 



( 25 ) sin (9o ( <w + </)'o ) [ 01, - 6 cos ^^ (o) + </>'o) ] = O. 



This equation may be satisfied in three ways. 



