n8 



of the computed mean anomaly is of the order of the firft power of 

 the excentricity. Therefore five or fix operations may be neccffary for 

 Mercury or Pallas, and in orbits ftill more excentric a greater num- 

 ber. Nor fhould we gain much in point of brevity, by previoufly 

 computing by Ward's or Boulliald's hypothefis, or even by Simpfon's 

 praftical rule, the true anomaly, nearly for the firfl alTumption. It mufl 

 therefore be concluded that although this rule is very convenient and 

 fimple in practice, yet it yields to Caffini's method, even when 

 correfted by Kepler's method. Both thefe methods in orbits of great 

 excentricity converge flowly. The following praftical method is free 

 from that inconvenience. 



A convenient praSlical Method of Solution. 



It has been fhewn that the error of Caffini's firfl approximation de- 

 pends only on the third and higher powers of the excentricity. So 

 that in orbits of fmall excentricity, the firft approximated is very near 



the true excentric anomaly. If c be the error of this approximation and c 



the approximated excentric anomaly itfelf, alfo m the mean anomaly 

 computed with the excentric anomaly c ; then as has been fhewn 



m 



nearly and the error of c thus computed is nearly 



i+ecs, c 



I 



i+ecSfC 



•therefore c depending only on the third power of the excentri- 



city 



