I 



26 



The preceding example Is not given as one that can occur ir. 

 praftice, or as the fliorteft method, in this cafe, * of finding the ex- 

 centric anomaly, but merely to fliew, that in the extreme cafe of ex. 

 centricity the method holds. It only fails, as was obferved, when the e!- 

 lipfe is nearly evanefcent, and at the fame time the mean anomaly nearly 

 1 80°. Machin's method gives, in that instance, the excentric anomaly 

 with very great precillon. But, as was before remarked, in very excen- 

 tric orbits, when the body is near perihelion, it is very inconvenient to in- 

 veftigate the true anomaly by means of the excentric anomaly, becaufe a 

 fmall error in the excentric anomaly occafions a great error in the true ano- 

 maly. And, although Machin's method furniflies us with the excentric 

 anomaly, in that cafe, with as great a degree of accuracy as can be de- 

 fired, yet the common tables of logarithms are not extended to a fufEcient 

 number of places to ufe them for that purpofc. 



The bell method of folving this cafe, which is that of the planets when 

 vifible to us, is to deduce the true anomaly meafured from perihelion, from 

 the correfponding anomaly in a parabola, having the fame perihelion dif- 

 tance. 



I had not Intended to notice this method farther ; but as the following 

 folution appears to be as fimple as can be defired, alfo readily admits of 

 being extended to any degree of accuracy, and at the fame time affords 

 another ftriking illuftration of the convenience of the method of deriving 

 feries above alluded to, I have been induced to add it. T. Simpfon and 

 Laplace are the only authors I have met with who have folved this problem.! 

 Their folutions give only the firft term of the correction to be applied to the 

 anomaly in the parabola. In the following, two terms of the correftion 

 are given, and more might be eafily deduced, if neceffary. 



Prob. Having given the anomaly in a parabola reckoned from perihelion, 

 to deduce from thence the correfponding anomaly in an excentric ellipfe, hav- 



* In and near the cafe of extreme excentricity, ihe common tables of natural 

 fines and circular arcs will readily furnifli, by mere infpe>.'lion, the eccentric anomaly 

 true to the nearell degree ; and then two operations will find it true to a fecond. 



t P. 58, 60. Mifc. Trafls. iVIecanique celefte, Tom. 1. p. 183, 1S6. 



ing 



