Of KEPLER'S PROBLEM. 231 



eular cafe, from 'which we can directly deduce a nearer value of 

 e than the fuppofition of e zz s, we will avail ourfelves of fuch 

 circumftances, and will thereby obtain a feries of approxima- 

 tions, converging fader to the eccentric anomaly. 



Resume the general equation of the arches of eccentric and 

 mean anomalies (Art. 2.) viz. 



m — {a — 1 fin p, : 

 it is evident, that the lefs s is, the lefs will be the difference of 

 the two arches m and p ; and that, when 1 is fmall, the quanti- 

 ty s fin ft, will be nearly equal to the quantity e fin m. There- 

 fore, if we take an arch r, fuch that 2r rr s fin in, it is ob- 



fin m ~ ** 

 vious that we mail have iELT — - nearly, and, confe- 



quently, e = s X - nearly. To fpeak more correctly, the 



error of the arTumption e z= g, will be of the fame order with 

 the third power of the eccentricity ; but the error of the afTump- 



fin 



tion, e — z - t will be of the fame order with the fourth 



r 



power of the eccentricity *. 



Gg2 It 



* Since m — /x. = t fin yt,, we have, in feries, 



fin m — f - ! ** m m _* E * fin s |t t _3_ 6 5 fin s ix ^ 



2 26' a 3 40 2 5 



Therefore — = 1 — ~ ■ « n * ** + %7Z fin V — & c - 



- («z — ^) 



- 07. (A 



and e = e X — = e — L_ fin * /* + 4r~ fin 4 ^ — &c. 



fo that the error of the fuppofition e = s, is manifeftly of the order eK 



Again, from the equation m — j« = £ fin u, we eafily derive fin p = 



f m w _ i fin 2777, neglefting the terms above the firft order : and, iubftituting 



2 



this 



