144 



more induced to do it, as thereby ftriking examples are afforded of the 

 advantage of the method of deducing feries which I gave in a 

 memoir read at the Academy, Nov. 1798.* The utility of this en- 

 enquiry will be more readily allowed at the prefent time, from the re- 

 markable circumftance, the difcovery of two new planets. Heretofore 

 an enquiry of this kind might have appeared of little ufe, as tables for 

 the equation of the centre were already conftrufted for all the planets. 

 Now we aftually have two new bodies, one moving in an orbit more 

 excentric than that of Mercury, for which the application of Kepler's 

 problem will be continually neceflary, till the elements of their orbits 

 are fettled with precifion, and tables contoured. It alfo may reafon- 

 ably be expefted that the induftry of aflronomers will, ere long, difco- 

 ver other bodies of the fame kind. 



The indirect method, which I have here recommended, and which for 

 orbits not very excentric appears to me as ready in praftice as can be defired, 

 is deduced from a combination of the methods of Kepler, Newton, and 

 the fecond Caffini; it is alfo applicable to the excentric orbits of comets, 

 and will, in all cafes, rapidly approximate. 



The mode of examination of the principal folutions, and the refults 

 from that examination are briefly as follows. 



The two feries for the true and the excentric anomaly afcending 

 by the powers of the excentricity, and by the feries of multiples of the 

 mean anomaly are firft given to ferve as it were for a fcale to meafure 



the 



* This Memoir was read at the Academy, Nov. 1798, and printed in the feventh 

 Tolume, publifhed in 1 800. In the latter year, a work of confiderable magnitude in 

 quarto, by M. Arbogaft appeared, entitled " Calcul des Derivations". The purport 

 of this work is precifely the fame as that of my memoir. The method of M. Arbogaft 

 is, however, very different from mine ; but, if I am not miftaken, my method is fuf- 

 ceptible of much more general application, and of greater facility in praftice. The li- 

 mited nature of a memoir prevented me from entering into much detail. I have 

 therefore engaged in drawing up a full and regular account of the method, and of its 

 application to all the purpofes to which M. Arbogaft has applied his, fome of which 

 had not fuggefted themfelves, till I faw his book, and to other important ones in which 

 his method does not appear to be readily applicable. 



