Chessin — On the Motion of Gyroscopes. 23 



tical diameter D^ with a constant angular velocity (o. 

 The direction Oil 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' ; OB is a fixed line in the equatorial 

 plane of the torus. The angles 0, (f>, 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 iVTV'. The angles which OP, OQ 

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

 ard y^; the principal central moments of inertia of the torus 

 (including its physical axis) about OP, OQ and 0]SrhyA,A 

 and C respectively ; A^, A^, C^, 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 (^' + ft)C0S7)2 + 6((/)' sin 6' + ft) C0S7J2 

 + C (1/^' + </>' cos 6 -\- (o cos 72)^ + ft)2 cos^ 7 

 + d (<^' + ft) sin fiy. 



where fi is the angle of OH with X, and 



(^^ {:: 



= A + Ai, b = A-\-C\ — A^, 

 = A, — A,— C„ d = A^-{- A,. 



\ cos 7 = cos /i cos 4> 



( 3 ) ^ cos 7j = sin /* sin 6 — cos /x cos 6 sin 4> ' 



I cos 72 = sin fi cos + cos ft sin sin ^. 



The differential equations of motion are given by the 

 formula 



d(3T\ 9T ^ ^ , 



We obtain immediately two integrals, namely 

 ( 5 ) i/r' -|- <^' cos ■\- 0) cos 72 = ^1, 



