VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 19 



It is evident that in this case the constraint is equivalent to a couple, whose axis coincides 

 with OB, let this be denoted by G. Then the equations (£) in the first section applied to the 

 case before us give 



— (Jw) = 5 — . Q sin - 5Q sin . — = 0, 

 dt^ ^ dt dt 



-(BQs\ne) = G + Aw.-'-B—.Qcose, 



at dt dt 



t 



-{B — ] =5Qsia0.12cos0-^ft,.Qsin0; 

 dt\. dt) 



from the first equation, w is constant, and from the last 



d"e (A 



«c' (A A ■ ^ 



-— = — — a> - Q cos y Q sin ; 

 dt' \B ) 



now in this case Q the velocity of the earth's rotation is very small compared with w, neglecting 

 therefore the second term of this equation, 



— - = — — (oQ sm Q, 

 df B ' 



whence the motion of the axis OA is precisely similar to that of the circular pendulum, whose 



S A Bs 



length is I, where - = — wO, and therefore 1= — , the direction of the earth's axis taking the 



I B AtoQ, 



place of the direction of the force of gravity. 



Also since-— = 0, when sin = 0, there are two positions of equilibrium of the axis OA, 



namely, when 9 = 0, and 6 = ir: the former is stable and the latter unstable, when w and Q 

 have the same sign. Hence the axis of rotation will remain at rest, if originally placed in the 

 direction of the earth's axis, stably or unstably according as the rotation regarded from the end 

 directed to the north pole is in the same direction, or the contrary, with the earth's rotation re- 

 garded from the same pole. If placed originally in any other position, it will oscillate about 

 its position of stable equilibrium according to the same laws as a circular pendulum. 



36. Next, let the polar axis OA always remain in the horizontal plane, and let (p denote 

 its azimuth from the south towards the east. Taking OB and OC as before, the latter will 

 now coincide with the vertical. 



If c denote the co-latitude, Q may be resolved into Q cos c vertical and Q sin c horizontal 

 in the north direction : hence the angular velocities by which the axes move, are relatively 

 to OA, OB, OC respectively 



. . d(b 



— Q sin c cos 0, - Q sin c sin 0, -^ + Q cos c, 



etc 



and the corresponding angular momenta are 



Am, - BQ sin c sin 0, 5 ( -^ + Q cos c j , 



3 — 2 



