110 



RELATIONS BETWEEN SEVERAL 



[BouK 1. 



It will be sufficient for our purpose to stop at the two first formulas. 

 By the first formula we have in our problem, 



if we put 



w). 



Wherefore, the first approximate value of \] p, which we will put = 3 a, will be 



, Arr 



\J p = j - 3a. 

 A; t cos 2 w 



By the second formula we have more exactly 



denoting by R the radius vector corresponding to the middle anomaly 



Now expressing p by means of r, R, r, N, N-\- i //, 

 inula given in article 82, we find 



4 sin 2 A sin \ A 



according to the for 



_ 

 P 



and hence 



cos^A __ , /_!_ , 1_\ _ 2 sin 2 ^ J _ cos to 



-B * \ r &quot; ~ 7/~ j9 



By putting, therefore, 



2 sin ^ ^f 

 p 



cos eu 



we have 



P _ cos ^ A \/ (r / cos 2 eu) 

 ~~ ^ 



cos w (1 -- ) 

 p 1 



whence is obtained the second approximate value of ^ p, 



