Of KEPLER'S PROBLEM. 229 



of Kepler's problem. Indeed, it is only this particular cafe of 

 the general problem, that we can confider to have been refolved, 

 hitherto, in a fatisfactory manner. We already pofTefs many 

 excellent folutions of this cafe, fome of them deduced from the 

 mofl elementary principles, and others obtained by the aid of 

 the higher calculus. To thefe another may be added, derived 

 as a corollary from the general folution contained in this paper, 

 and which will be found not unworthy of the notice of aftro- 

 nomers. 



Because the two arches denoted by /& and p are the double 

 of the arches v and t, the arch p — />, which is the error of />, 

 confidered as an approximation to the eccentric anomaly, will 

 be double of the arch v — v : therefore, it is obvious, (Art. 8.) 

 that the arch p — p will never be greater than 



(f — tan $ */ 1 — ijjl 



r 



taking tan § — V 





The flightefl attention to the nature of this expreflion, is fuf- 

 ficient to evince, that it decreafes very rapidly as s decreafes. 

 If we evolve it into a feries, proceeding according to the powers 

 of e, that feries will contain only the third and higher odd 

 powers. Therefore, when t is fmall, as it is in the cafe of the 

 planets, the amount of the above formula will be inconfiderable. 

 It may even be fo inconfiderable, that the error of p will be of no 

 account in practice, and the firfl approximation will give the 

 eccentric anomaly with the requiute exactnefs. 



By means of this formula, I have computed the limit of the 

 error of p, for all the planetary orbits, and have arranged the 

 remits in the following table : 



Vol. V— P. II. G g 



