38. 



ON NEWTON'S SOLUTION OF KEPLER'S PROBLEM. 



[From the Monthly Notices of the Royal Astronomical Society, Vol. XLIII. (1882).] 



OF all the methods which have been proposed for the solution of this 

 problem, that which leads most rapidly to a result having any required 

 degree of precision may be briefly explained as follows : 



The equation to be solved by successive approximations is 



x e sin x = z, 



where z is the known mean anomaly, e the eccentricity, and x the eccentric 

 anomaly to be determined. 



Suppose to be an approximate value of x, found whether by esti- 

 mation, by graphical construction, or by a previous rough calculation, and let 



Then if & " Zl 



- 



1 e cos x 



and x f =x (l + 8x , 



x f will be a much more approximate value of x than # . 

 Similarly, if we put 



and if 



x' e sin x 1 = z', 



z z' 



- r 



le cos x 



A. 37 



