103 



// / 



c — c 



Newton's firfl approximation is c= ^ or rather 



^ i+ecs, c 



c :f__£ ::_ : — i-"c,aswillappearbyareferencetolusruIeandif infleadof 





c, c+c thus computed be ufed, and then c=m—es,c+c and c 



be computed c+c+c will be a nearer approximation to c, which may be 



continued at pleafure. 



With refpeft to the rate of convergency of the quantities 



\ 

 \ 



c+c, c+c+c, &c. to c. 



\ \ w 



w \ \ 



c — c [ecsc. 



We have c = x i Now if the error of c be of the fame or- 



\ 



i+ecsc i+ecsc 

 der as the firfl: power of the excentricity, the quantity c—c may be of the 



c — c \ecsc 



fame order and .-. the error in c ariCng from neglefting x Jn^y 



i+eac i+ecsc 

 be of the order of the third power, becaufe c and e are of the firfl; power. 



Hence affuming the excentric anomaly = the mean anomaly, the error 

 after the firft Newtonian operation cannot be of a higher order than the 

 third pov/er of the excentricity, becaufe the difference between the mean and 

 excentric anomaly = e s, c. By the fame reafoning, if the excentric 



anomaly 



