80 PHILOSOPHICAL TRANSACTIONS. [aNNO 1791. 



As ; — r— T =s — , + - — - -1- &c. ad inf. ; we have, by the same me- 



X X (x + 1) X^ 1^ x' x^ ' ' •' 



thod of proceeding, a — b + c — d+ &c. ad inf. = ±; consequently a -|- c 

 + E + &c. = f , and B + D + P + &c. = J-, 



Because , — =r - + -, H — ; + &c. ad inf.; if for x we write 2, 4, 6, 



(.r — 1 ) X X x' ' j' ' ,r' ' , J J 



&c. then will JL + JL 4. J_ + &c. = (tab. 3) a" + b" + c" + d" + &c. ; 



X«x OiT* J*" 



but -i; + — + -i-g + &c. = hyp. log. -2; hence a" + e" + c" + d" + &c. 

 = hyp. log. 2. 



If in the same expression we write 3, 5, 7, &c. for x, then — + t—; + g-i 



+ &c. = (tab. 4) a" + b" + c" + &c.; but ^ + j^ + 5^ + &c. = 1 - 



hyp. log. 2; hence a" + ^" + c* + &c. = 1 — hyp. log. 2. — Hence from either 

 of these two last cases, we have a very expeditious method of finding the hyp. 

 log. 2. 



Prop. 2. — To find the sum of the iv finite series whose general term is — —r— . 

 ^ ^ ^ mx' ± n 



By division — — ;-— = — - + — ,-„- + -r^ + —rr. + &c. ad inf. ; hence, if ,. - 



be made the general term of a series, and for x we write 2, 3, 4, &c., its sum 

 will be equal to the sums of another set of serieses, whose terms are the 

 powers of the reciprocals of the natural numbers respectively multiplied into 

 — , —C-, — o &c.; hence the sum of each of these series being known from the 

 tables, the sum of the given series will be found. 



Exam. 1. Let - — be the general term; now „ , . = — :+-» j + 



x" -^ X ° j;' + 1 .r- x* j" x^ 



&c.; hence if for x we write 2, 3, 4, &c. we have 1+^+7^ + 57: + ^c. = 

 A — c + E — G + &c. = (by tab. 1) .57667 4037469. 



Exam.1. Let-^ be the general term; then, by the same method of pro- 

 ceeding, ^ + J + -j^ + i^ + &c. = a + c + E + &c, = (by prop, l) 



3 

 V 



1 , 1 



Cor. Because i + 1 + 1 -f &c. = i X ( 1 + J + J. + &c.) = (as 1 + ^ + ^ 



-f &c. is the reciprocal of the figurative numbers of the 2d order) - x 2 = -; 



therefore i + 1 + 1 + &c. = J. Also, as ^-i- = 1 + i + -, + &c.; if 

 3 ' 15 35 ' 2 j:' — 1 x^ x'^ ' x^ 



we write 2, 4, 6, &c. for x, we have - + 7; + „-t + &c. = (by tab. 3) a" + c* 

 + e" + &c. = ^; but, by prop. 1, a" + b" + c" + d" + &c. = hyp. log. 2; 

 hence b" + d" + f'' + &c. - - ^ + hyp. log. 2. 



