354 Astronomical and Nautical Collections. 



And in a similar manner we find the co-efficients of the other 



series I - 1.3.5 . . (2n - l)«ill + (»-l) in-1)xx ^ 

 ^ ^ x'"+' 2(2n-l) ,z 



(tz — 3) (w— 4) XX g , {n — 5) (n — 6) ar.r ^, , 



4(2«-3) zz 6(2k-5) zz 



Moreover, if ??« be the distance of any term of the series of suc- 

 cessive y?wenfs, s, s', s", from the term s, writing — m instead of ?j, 

 we shall find the value of s'" by the same series. In this case, we 

 must still take the co-efficient of the first term, such that its greatest 

 factor shall be 2ji — 1 == — 2m — 1. But the co-efficient 

 1.3.5.. (2?i — 1), or (2?i — 1)..5.3.1. may also be written 

 (2m-1)..5.3.1.-1.-3-5. .. _(-2m-l) (-2?w-3)(-2m-5)... 



_1._3._5... _1._3._5... 



Now since n is here supposed to be negative, and m affirmative, 

 the whole of the factors {—2m - 1), (— 2m - 3), (-2m - 5). . . 

 in the numerator are taken away by the same factors in the deno- 

 minator, and there remain only ; so that 



_l._3._5..(_2m+l) 



«+i 



- 1. -3.-5. .(— 2m+ 1)0,--'"+' 2( — 2m — 1) 



._2,„+. 2r-2m — n zz 



( — 771 — 1) ( — ?n — 2) XX 



B. 



w.2m— I 



4( - 2m - 3) ZZ -].-3.-5..-(2m-l)z-^» 



( — m + l)m XX J. , (- 7/1 — 1) {in -f 2) jrx „ 

 2(2;« -f 1) "zz 4(2m-t3)zz 



(-ffl-3)(»z4-4] XX Q j^ . that j according to the former 

 6(2m -Y ^)tz 



series : and this series is the more convenient for finding the fluents 



s', s", s"'. . . ; the other for the fluxions s, j, . . . 



For completing the expressions, since r =r t/z, and y =: V^, we 



c 



find the fluents r = c?/, r' \y=i (cyz)' — [cr)'] ~ c-y, r" == c'>/. . .. 



and taking the fluxions, r = [j/z ;: ] y, t —L., r ■=. X, . . . 



c cc 



