9° 



The laft indirect method which I have noticed, and I know of none 

 of any confequence later, is De la Caille's.* Lalande recommends 

 this in praftice, and has himfelf given a demonftration of it; 

 But it does not appear to be fo convenient in praftice, as De 

 la Caille's improvement of Caffmi's folution, except when the true ano- 

 maly is very nearly known ; for, as is hereafter fhewn, the refpeftive 

 correftions proceed according to the cxcentricity, and confequently in 

 excentric orbits, mufl be repeated many times. Indeed, in examining 

 the accuracy of a table of equations of the centre, this method is 

 very convenient. 



I am aware that there are other folutions of this problem, not ex- 

 amined here, fome of which did not require notice, and others, as 

 Lorgna's and Trembley's, cited by Montucla,t I have not feen, but fup- 

 pofe if they had fumifhed any confiderable improvement they would 

 have been detailed in the laft edition of Lalande's Aftronomy. 



Of the Series for exprefflng the excentric and true anomaly in terms of the 



mean anomaly. 



Let m = the mean anomaly to radius unity, e = the cxcentricity of 

 the orbit, the femiaxis major being alfo unity, and let c and a repre- 

 fent the excentric and true anomalies refpeftively, then 



c=m — es, m+e s, 2m — e . y, ^nt^-Sjm +e i s, 4m — s, 2m+ k<u ■ 

 1.2 1.2.4 1.2.3 



* Mem. Acad. 1750. Mem. Acad. 1755. Lelande's Ed. Hall Tab. 

 t Hid. Mathem. vol. 2. p. 345- 



