Of KEPLER'S PROBLEM. 231 



C"ular cafe, from -which we can direcflly deduce a nearer value of 

 e than the fuppofition of ^ = e, we will avail ourfelves of fuch 

 circumflances, and will' thereby obtain a feries of approxima- 

 tions, converging fafter to the eccentric anomaly. 



Resume the general equation of the arches of eccentric and 

 mean anomalies (Art. 2.) viz. 



m — ^ = s fin fi, : 

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

 the two arches m and ^0, ; and that, when i is fmall, the quanti- 

 ty £ fin [/^ will be nearly equal to the quantity s fin m. There- 

 fore, if we take an arch r, fuch that 2r = f fin -•», it is ob- 



fin ^^7^ 

 vious that we fliall have -ilLT — nearly, and, confe- 



- {"' — f^) 



fin 



quently, e zz i X — - nearly. To fpeak more corredly, the 



error of the afl!umption f zi e, will be of the fame order with 

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



r 



tion, e ~ i ^^^, will be of the fame order with the fourth 



r 



power of the eccentricity *. 



Gg2 It 



* Since m — /* = e fin /«, we have, in feries, 



fin "^ — 1^ E fin m \ e' fin ' m _3^ _ £ ' fin ' ^ ^^^ 



2 2 62' 40 2' 



£»! — u, 

 n fj 5 £* 



Therefore ^— = ^ " i^ • «" ' ** + g^ fi"' '* - ^^• 



r m — I* 



— 2 5 ^ E ' 



and f = £ X = £ — !_ fin ' A* + ^ fin * |i^ — &c. 



2 

 fo that the error of the fuppofition e = e, is manifeftly of the order e'. 



Again, from the equation »j — ,« = e fin a, we eafily derive fin ^ = 



^mm — - fin 2w, negkaing the terms .above the firft order : and, lubftituting 



i" u Hi \(i;ri:i'}n -Ji\. ..y 9iol3i:i.. this 



