356.] BODY WITH FIXED POINT. ! 9 5 



To determine the constants Ci, C. 2) the initial values of o^, <o 2 , say at 

 the time /= o, should be known. Let e be the angle made at this time 

 by a> with the principal axis b. Then the initial values of o^, o> 2 are 

 w sine, eo cos e, and the substitution of /= o in (13) and (14) shows 

 that Cj = w sin e, C 2 = w cos e. Hence we have, finally, 



coj = w sin (Awg/ 1 + e) , o> 2 = o> cos (A.<u 3 /+ e), o> 3 = const. (15) 



355. To determine the position of the body at any time / with 

 respect to fixed axes through <9, let us take as axis of z the fixed 

 direction of H, which is perpendicular to the invariable plane. The 

 cosines of the angles made by this axis with the principal axes are 

 found similarly as in Art. 335 (see Fig. 42) : 



cos ZX'= sin < sin 0, cos ZY' = cos < sin 0, cosZZ'=cosO. 



Hence the components ff 1 =7 1 a> lf H 2 =I 2 w 2 , J^ 3 =I 3 (a 3 of H along the 

 principal axes are 



/id)! =7/sin < sin 0, 7 2 o> 2 =ZTcos <f> sin 0, ^wg =^ffcos 6. 

 These equations give 



cos0=^, (16) 



and, as 7j =/ 2 , tan </> = = tan (A<o 3 /+e), 



o> 2 



by (15) ; hence <#> = X^t + ^ (17) 



where ^> is the value of < for / = o. Thus it appears that the angle 6 is 

 constant, while < increases proportionally to the time. 



356. To find ^, we may use the third of the equations (5), Art. 335, 

 viz. o) 3 = < + \j/ cos 0. As cos = 1^/11, <j> A<o 3 , A = (7j 

 we find 



whence 



dt 



TT 



It appears then that the equatorial plane X' Y 1 of the body remains 

 at a constant inclination to the invariable plane, while the nodal line 

 ON (Fig. 42) turns uniformly in this invariable plane and a radius of 

 the body in the equatorial plane turns also uniformly in the equatorial 

 plane. 



