96] PRECESSION AND NUTATION. 301 



so that, inserting in 163), 



jj = 7 L_ 1 sm 2 1 sm ft, cos ^ 

 170) 



Ti/r 3ym(A C} . ^ 7 . 

 M = ~ ' sin 6 cos Z sin &. 



If we suppose the sun's path relative to the earth to be a circle, 

 described with angular velocity n, we have 



so that 

 171) 



d L &yin (C A) \ 



3ym'(AC) f 07 . . 7 ^d& 



r s" I*' 1 COS ^ Sm ^ + sm * cos ^ COS # -TT 



dt 



Now if A = C, there would be no motion of the earth's axis, so 

 that C A is a small quantity of the order of -=J The angular 

 velocity n, though much larger, is still 365^- times smaller than i&, 

 so that if we neglect its product and that of ^- with C A, we 

 may neglect the right hand sides of 169). Thus the approximation 168) 

 is justified, for differentiating, it will make ~ > -jrf negligible, so 



that equations 167) are satisfied. Inserting the values of p, q, L, M, 

 in 168), we have 



(C ~ ^cosfrd -cos2r>, 



These are the equations for the precession and nutation. In order 

 to integrate them approximately, we may neglect the small difference, 

 on the right, between # and its mean value, so that inserting the 

 value of 21 = 2nt -\- 21 , considering & constant, and integrating, 



3ym C - A _, 



* - ifir* "(T COB *' " 



J 3ym C-J. ,, 



* = sm * 



We thus find the motion to be a regular precession, of amount 



t rr ,j\ i 3ym (7 ^4. 



174 ) * =db^r- cos *' 



together with a nutation in an ellipse (compare 93), whose period 

 is one -half that of the revolution of the disturbing body. 



