Chessin — On the Motion of Gyroscopes. 23 



tical diameter DU with a constant angular velocity a>. 

 The direction 0X1 is that of the positive axis of rotation of 

 (Cg). The axis X is the projection of OH on a plane per- 

 pendicular to ZZ ' ; OH is a fixed line in the equatorial 

 plane of the torus. The angles 0, (j>, yjr are Euler's angles 

 defining the relative position of the body with regard to 

 the moving coordinate system XYZ. The axis QQ' is per- 

 pendicular to PP' and NN'. The angles which OP, OQ 

 and ON form respectively with OH will be denoted by 7, 7j 

 ard 72 ; the principal central moments of inertia of the torus 

 (including its physical axis) about OP, OQ and OiVby^, ^ 

 and C respectively ; A^, A^, 6\, and A^, A^, C^ will indicate 

 the principal central moments of inertia of the ring (C^) 

 about OP ', ON, OQ and of the ring ( C,) about OP ', OZ, 

 Oq. If, then, T denote the kinetic energy of absolute rotary 

 motion of the gyroscope about the point ( O ) , we shall have 



(1) 2T = a (6' -{■ CO cos 7)2 + b (<t>' sin (9 + <u cos y^y 

 + C (y{r' + <f)' cos + CO cos 72)^ + 0)2 cos^ 7 

 -f- d (<f)' -\- CO sin fiy. 



where /x is the angle of Oli with X, and 



ja = A + Ai, b = A-\-C\-A„ 

 ^^ \c = A, — A,— C„ d = A, + A,. 



I cos 7 = cos fj. cos ^ 



( 3 ) < cos 7j = sin ft sin ^ — cos fi cos sin <f> 



I cos 7, = sin fi cos + cos (m sin sin ^. 



The differential equations of motion are given by the 

 formula 



We obtain immediately two integrals, namely 

 ( 5 ) 1^' + <^' cos ^ + w cos 72 = ?i, 



