86 THILOSOPHICAL TRANSACTIONS. [aNNO 1791' 



1^0.-1 -I-'-- &c. Now, by division, ^-^— = i-^ + ^-^.+ 

 &c.; hence, by writing 2, 3, 4, &c. for x, we have (after transposition) - 

 + 1^ + &c. — ^ — ]■ — ^ — &c. = — iA + 4-B — x c + &c.; hence, by 

 adding equal quantities to each side, we have ^ + J + J + ^^- ~ ^ ~ 3 ~ 

 - — &c, = ~ — J-A + 4-B — ic + &c.; consequently 4-a — ^b + -fc — &c. 

 = -—-—-— '-— &c. 4- 1 + -+ ]r+ &c. = ^ — hyp. log. 4. 



Prop. 14. To find the sum of the infinite series ^— + — - + ^— - + &c. 



The general term, by writing 2, 3, 4, &c. for x, is ^^^J^ ^^ = 2? ~ JT^ + 

 -i; — &c.; hence the sum = 4^A — -I^b + ic — &c. = (by prop. 13) -—hyp. log. 4. 



Prop. 15. To find the sum »/ ^ + j + J + ^"j- 



The hyp. log. ^— =] + ^. + ^ + ^^ + &c.; consequently hyp. log. 



J 1 1 1 



J - 1 sT' 3^3 47* 



we 



&c. = -; hence, if we write 2, 3, 4, &c. for *, 



have hyp. log. j + hyp. log. | + &c hyp. log. j^— _ 1 ( x i-,+ 



L + &c. .. ij - L ( X J;+ ;-+ &c. .. i) - 1- ( X ^- + ^-T+ &c... J) 



— &c. &c &c. = ^ + i- + J + &c J ; but hyp. log. j + hyp. log. | + 



hyp. log. J + &c hyp. log. j^ = hyp. log. } X^ X*- X &c 



11 1 



— ^ = hyp. log. x; also ^1 + 3^ + &c - = the sum of the same series 



ad infinitum, minus the sum of all the terms from - = (if x + 1 = n) a 



L _ _L -I . 4. &c.; in the same manner „, + - + &c - 



= B :+ — . r-o + &c.; and so on for the other serieses; 



2n' 2n* 4«-' ' ]2n° 12/i'* ' ' 



hence, by substitution, and adding unity to each side, we have hyp. log. x + I 



— ^A — iB — 4c — &c. + - + j,,^^ + ^^, -i- j,,Q^, -\- ^^, -\- 25,^,„, + J^^^ + 



^^, + &c. = ] + i + i + i + &c. . . . i; but 1 - iA - iB - ic - &c. 



= .577215664901 ; hence 1 + i. + 4- + &c j = hyp. log. x + 



.577215664901 +±+ j±^ + ;^^ + £-^ +J^ + ^L + J.+-.^l_ + &,. 



Prop. i6. To find the xalue of a. XPXyX^X &c. ad infinitum, sup- 

 posing the general term to be a rational function of x. 



