12 PRECESSION OF THE EQUINOXES AND NUTATION 



the rates of precession will be 



(') K(P+ n 3 P'+2n 3 P') for nj= 



(ft') K(P 7* 3 P'-j-2M 3 P") " 3 <= 7t 



(c'j K(P-2n,P") " M 3 <=j7t 



Equating (a) to (a'), &c., we deduce 



P=\ cos /(cos 2 /'+cos 2/0=cos /(I 2 sin2 ^O 



2% sin / ; 



P"= - cos / (cos /'cos 2/0= -r3 cos / sin 2 T. 

 4 3 4 



Hence the formula for precession may be written 



44. -^4^ ^^ cos 7 f( 1 I sin2/ ') ^ sin 2r cot 2/sin "3 



2.n(l -}->7) v L v 2 w 3 



a 



sin 2 .T sin 



Similarly Ave would get for the nutation 



A.. 3 n<? C A T sin T 



4i3. ' --- COS / - - COS 3 <. 



The ratio of actual lunar precession to what it would be were the moon's orbit 

 in the ecliptic, is therefore expressed by 



1 sin 2 r^O.99 (very nearly). 1 



t 



The third term of (44) indicates a slight periodical variation from the true 

 elliptic motion referred to in the next paragraph. There should be a corresponding 

 term in (45) which may be obtained by the same process, but they are both too 

 minute to enter into computations. 



1 It is worthy of remark that the formulae of Laplace [3100] and [3101] (Bowditeh) contain no 

 such coefficient qualifying the mean lunar precession, though one is found in all the more popular 

 solutions ; neither do they contain the term (quite minute) in 2 3 < of (44), but, on the other hand, 

 contain terms in 2n a i> (corresponding to the terms in 2n,< of 25 and 26), which, referred to in the 

 fourth par. (page 9), are generally omitted as inappreciable. 



