444 PHILOSOPHICAL TRANSACTIONS. [aNNO J 76O. 



Hyp. log. of - = 1 — J7 + ^ — 2 ^ ^ — 3 — ^ ^^* ^'^^ consequently x = — 



(1— x)» (1 — x)' o 



J - X - ^^ - ^-V-, &c. 



Moreover the fluent of - x hyp. log. of j—— is = a: + - + -, &c. which 



vanishes when x vanishes ; and the fluent of . Xxis=: (i — x) + iinfi.* 



I — X ^ '^2* 



/J j'\ 3 



4- ^s — j-^, &c. — p", being corrected so as to vanish when x vanishes. 



But the fluent of - X hyp. log. of- 1- fluent of —^ — X x is = x X 



hyp. log. of , which also vanishes when x vanishes. 



Therefore x X hyp. log. of j~ is = ^ + i + |^, &c. + 1 - or + ^^ 



, (1 — X)3 Q „ 



4- i , &c. — f". 



Whence, by taking oc equal to i, we find — square of hyp. log. of 2 = 2 X 



(t4^ "^ 2^2"* + 3^2^' ^^'^ ■" ^"' ^^"^^' ^" ^^^"^ ^^^^^^ ^^^^^ - T' ^^ ^PP^rs 

 that when x is = 4, the series ^ + |a + |„ &c. is = ^ — 4. x (hyp. log. 

 of 2y. 



X* x' 



7. Further, jc + ^ + Jjj &c. the value of f" in art. 2, must be equal to 



Tf 3 J a X~"' 



2p''x + ix^ — T-r + 57-' + — 5- + -Tj-j &c. the value of p'^ in art. 5, when 

 both series converge. 



Therefore ^-^^' + — "=r'- + ^^'. &c. is then = 2p'^x + ^x^ - ^. 



Whence, by taking x equal to — 1, we have Ab^?" + 46' — — - = o ; and, 



consequently, v" = -|-, as before found. 



And, by taking x equal to ^"J^ we 



find 



-jP= X a - 26P i- ^ 2.3 — 3V— 1 VZl + 2.3VZT - 2^=1* 

 Therefore a " is = ^ . 



8. From what is done above, it evidently follows, that 

 i^ . 26»p" , 2.86* 



vi __ 26^?^' 86^p" 3.326" 



&c. &c. 



&c. , &c. 



