[ 348 ] 



Let the fucceflive fluxions of this equation be taken by Prob. 2 



and 3, when e — 0, and c :=,m 



c jf. £ s, m = 



c + zeccs, m = o 



i 



c + ^ec cs, m — 3 ec"^ s,m=:o 



4 . ^ . . ? 



c + /\.ec^ cs, m — 4 e. ^ecs, m — 4 ec cs, m — o 



whence fubftituting for r, e 



e = — e s, m.c = — tcccsym = 2^ s,mxcs,7n=^ e^s, im 



3 » 



f = — 3 <?* c cj, w + 3 f C2 J-, iw = _ 3 f 3 X J-, 2mx cs, m — s'i, m = 



— -I f ' X 3 J"> 3 >"— -Ti iW' 



dr = — 4 d" X c rr, w — 3 c CJ-, »z — c CJ-, »< =r 4 f 4 X 2 /, 4 w — J", 2 w 

 &c. &c. 



-.• c = m Jr-c -\ + &c. — m — es^m A j, 2 w- — 



^ ^ 2 1.2 I. 



2. 4 



<:* 



3 J-, 3 wz — X, w ^ >< 2 /, 4 m—s, %mj^ &c. • 



This feries is in efted the fame as the fcries given by Keil, but 

 is much better adapted for computation, and befides has the ad- 

 vantage of being apphcable to phyfical aftronomy ; which the feries 



of Keil is not.* 



Example 



M. De la Grange has given a moft elegant theorem for Cxprefllng in a feries 

 afcendlng by the powers of t any funftion of x, when x z= any funftion of » -|- /X, 

 X being a funftion of x. By h^lp of his beautiful theorem, the value of c is 

 immediately deduced from the equation m =: c ■^- e s, c. Bat as the theorem is only- 

 adapted to equations of that particular form, it appears equally eligible to deduce the 

 value of c by the above method, becaufe including the demonftration the method of 

 De la Grange is not ihorter. See Coufin's Aftro. Phyf. Art. 20, page 15. 



