no 



n= ^ 



5 



2-3 2.3.4.J 2.3 .. 7 



« 1 3 « 1.71 3 5 K !...« 5 7 



eiiK : — jH ^ s + x &c. 



2.3 2.3.4.5 2.3 7 



If fX« — I . n — 3 =9, the CO efF. of x'=oand «=v/5+^9+i6 



now as e cannot be greater than unity, this value of n can never be lefs 



thauv/io and therefore with it J (=fine -) will be always fo fmall that the 



n 



above feries will fwiftly converge. 



The equation becomes taking this value of tu 



m 71 — I. e+i , n ■, 



— =1—6 s+ . s H s +&c. 



n 2.3 560 



neglecting the terms after s the value of s is had nearly by the refolution 

 of a cubic equation, and s being found na or c is thence computed for the 

 firft approximation. 



Had the motion been reckoned from aphelion, the value of 71 would 



havebeen^/j+v^g — 16, and therefore would not have been generally 



e 



polTible. 



To corre(rt this approximated excentric anomaly ad libitutn, Machin 

 computes by it the mean anomaly correfponding, by Kepler's rule, and 

 the error of the mean anomaly fo computed, he divides by the planets 

 diftance from the fun, the quotient is the correflion to be applied to 

 the approximated excentric anomaly, to obtain a fecond approxima- 

 tion, &c. 



The 



