VOL. LI.] PHILOSOPHICAL TRANSACTIONS. 443 



1 + J. + ii + f-,, &c. = q'', 



1 + ^4 + 5^ + p &C. = Q% 



&c. &c. 



5. Multiplying the last equation in art. 3, by -, and taking the correct fluents 

 we have 



f' = 2P^' + 2Zjx - ^ - a;- - ^ - ^', &c. 



Whence, by multiplying by -, and taking the fluents, we get 



P" = 2p"x + ^x^ - ^ + ar-' + ^ + ^', &c. 

 I 2 3 ' 2' ' 3^ 



Again, multiplying the last equation by -, and taking the correct fluents, we 

 find 



P =2p" + px^+____x----^, &c. 

 And by proceeding in the same manner, we find 



p. = 2P"x + £|i + *^ _ jj^^ + .- + g + g, &c. 



&c. &c. 



6. Now, it is obvious, that x •}- -^ -\- |-^, &c. the value of p' in art. 2, must 



be equal to 2p'' + "^^x — ■^~* — -^^ -^j &c. the value of f' in art. 5, 



when both series converge. 



Therefore, ^,— + -t,— + "-^^^i^* &c. is then = 2p^^ + 2/»x - -. 



P 2* 3 2 



From which equation, by taking x equal to — 1, we have -| — ^ — ^ 



+ ^2 — , &c. = y" -\- y^ =. -p" — a^ ; and, by taking .x equal to "7=^, we have 

 - JlH-^.-3- + ^.- &c. = 4p'' + 3P = 4P'/-3a^ ' 



Therefore 4p^'' — 3a^ is = p^'' — a^: hence p''' is found = ^—, 



3 



Moreover Ti + ^ + 02 + n^ ^c. being = p'', by supposition, and -— 

 i~ + 22 ~ T2 H~ Tj ~^ ^c- = ^'" — «^ as found above; we, by subtraction, 

 get fi + 3T + jiJ &c. (= 2a') = a% and, consequently a" = ""-. 



1 X* x' 



Scholium. The hyp. log. of — — - being = .r + ^ "^ "3' ^^* ^^^> ^J ^'rit- 

 ing 1 — X instead of ar, have 



3 l2 



