352 



Lastly, in ^'the unrestricted case," where both rings must be left 

 free to move, let the line round which the outer revolves be placed 

 parallel to the earth's axis. Including the rings in moving system in 

 this case, and applying as before the equations of relative ms viva and 

 relative moments, I have reduced the determination of the motion of the 

 axis to the following pair of equations : — 



d^lr Cn (cos Oq - cos 0) + wITq 



^ (1^) 



where ^= ^ sin + cos -^6 + Ci sin ^6 + A^. 



It will be at once seen that an exact solution to correspond with a solu- 

 tion of this case, when the rings are not included, is not to be hoped 

 for. It may, however, be readily shown that, to a very high degree of 

 approximation, the motion of the axis is still that of a retrograde rotation 

 (lo) round the polar line, combined with an infinitesimal conical nuta- 

 dO 



tion; for, equating — to cypher, and neglecting terms in (w^), the limit- 

 ing values of 9 will be found to be Oq and (^o - 2j?), where 



p 



Cn sin Oq 



Assuming 0 = its mean value [^o - !P\ +3/? and omitting terms of a 

 higher order than (^z), we get on substituting in (15) 



or writing 



Cn sin Oq 



q = 



y{A+A,)jiQ 



^JL = -qy^Zy ,j=.p COS (qf\ (17) 



the arbitrary constant vanishing, since ^=p when t = 0. 



, . d-^ Cwsin^o / ,x . ^ A ^ 



Again, + = — y = cos (qt), sm Oq l-^ + iv j 



• o / ,x 



— j, say = ic sm Oq cos {qt) ; 



