446 PHILOSOPHICAL TRANSACTIOXS. [anNO I76O. 



12. Now, it is obvious, that a? -|- .^^ + - , &c. the value of g' in art. 9, 



must be equal to 2a'' -\- bx — x~^ — -— — , &c. the value of g' in art. 1 1 , 



when both series converge. 



Therefore ii^ + ^-t^' + ^^^^', &c. is then = 2a^ + 6x. 



Whence, by taking x equal to ~r=^i we have 2q' -\- b^ = 0; and, conse 



quently, q^'' = — as in art. 6. g^^^ 



1 



X 

 X3 , X' 



13. Likewise a; + - + r-, &c. the value of o'' in art. 9, must be equal to 



2a''x + ^ + ^~' + V + ^' ^^- ^^^ ^^^^^ ^^ ^'' ^" ^^- ^^y when both se- 

 ries converge. 



Therefore ^^' + "^-^^ + '-=^-> &c. is then = 2q''x + *-|-\ 



Hence, by taking x = — =-, we find ^== X a'" = 2bQ.'' + g = sV"^ » 



and, consequently, a" = -, as in art. 7. 



14. From what is done in the last 3 articles, it evidently follows, that 



-ais = — + — 3, 



VZi — ^« T^ 2,3 T^ 2.2.3.4' 



_a« ='-!«:' + £5: 4. _»'!„ 



2 ' 2.3.4 ^ 2.2.3.4.5* 

 Q"' ^. 6'q"' b'Q" b^ 



V~l = ^<^ + "I".? + 2X4T5 "^ 21.3.4.5.6' 

 &C. &C. 



Whence (as well as from the theorems in art 8) may the values of a'% «% 

 a"', o""', &c. be readily found, in terms of a. 



15. g' being = j:' -f i| + ^, &c. by art. 9. 



h' = fluent of xia' is = ^-^ + ^3 + -rrj, &c. 

 h" = fluent of ^ H' = ^^ + 3^. + 5~, &c. 

 H- = fluent of xiH" = -^- + 3r^y + ^}r^, &c. 

 «' = fluent of ^h"'= -^J,-- + -^. + ^^ &c. 

 &c. &c. &c. 



16. Moreover, g' being = 2q" •{■ bx — x~' — *^ — t7» ^^* ^7 ^^t- 1 1» by 

 multiplying by x.v, and taking the correct fluents, we get h' = x^ik" — a^ -|- 



*4- - T + ^ - ^ + ^ + ^'' +i9 + 5^' + S* ^^- '" ^^^"g p"^ ^°^ t^e 



^"^ K3"« +5^5^ + sT?"** ^- 



