91 



Thefc fcries are here given to ferve for fcales of comparifon, as it 

 ■were, to the dJiferent feries hereafter inveftigated from the refpeftivc 

 hypothefes examined. It is fufficient for my prefent purpofe, merely 

 to flate the feries without entering particularly into their inveftigation. 

 They are to be met with in various authors.* I fliall only obferve that 

 the feries for c, is derived from the equation c=m-es,c. This equation is 

 readily folved, and alfo any funftion of c found by a remarkable and 

 elegant theorem invented by Lagrange,! and afterwards demonftrated 

 by Laplace. The latter theorem for the true anomaly has been invef- 

 tigated by feveral authors by reverfing the feries for expreffing the 

 mean anomaly in terms of the true. It has been deduced by others 

 from firfl finding the exceatric from the mean, and then the true from 

 the excentric anorraly. The latter mode is adopted by Laplace, 

 in his incomparable work " Mecanique celefte." This great mathe- 

 matician has there given an inveftigation flrikingly elegant. He firft 

 has obtained, by an ingenious transformation, the law of the feries 

 expreffing the true ia terms of the excentric anomaly and excentricity. 

 By combining this conclufion with the feries for the excentric anomaly, 

 and the finss of its multiples, the feries for the true anomaly may b« 

 continued at pleafure. 



It is hereafter pointed out, how the fame feries for the true ano- 

 maly, in terms of the excentric, may be obtained without the intro- 

 duftion of impoffible quantities. The law of the feries, indeed, is not 

 dcmonftrated, but only colle(fted by induAion^ yet it may be a quef- 



( M 2 ) tioft, 



* Lagrange Berl. Aft. 1769. Coufin's Aft. Phyf. 43, 44. Laplace Metanique 

 Celefte, Lit. 2. c. 3, 22. Lalande Aft. vol. 3. Tran. R. I. A. vol. 7. 347, ^^o. 



t Berl. Acad. 1768. 



