VOL. LXXXI.] PHILOSOPHICAL TRANSACTIONS. 85 



Here the general term is ^^ _ ^,^ . j3 _ ^^ ^ ^.^ = J^ + ^^^ + 7^ + ^^' ^"^ 

 hence the sum = h + 3k + 6m + &c. = .004707 M8337. 



Prop. 9. To find the sum of the infinite series 1 — 3 + g — 7^+^^- ^^'"o 

 n series of the recijjrocal of the figurative numbers of the 3d order, having the 

 signs alternately + and — . 



This series, by resolving two terms into 1, becomes ^ ^ ^ + — j— j + 



— t 1_ &c. whose general term, by writing 2, 4, 6, &c. for x, is 



5.0.7 



= -, -1 — ,+-.-{■ &c. consequen'Jv the sum = 4b" + An" 



(X -1) X X X (.x+ I) j;' ^ x' ^ I' ^ ^ - ^ 



+ 4f" + &c. = (by cor. ex. 2. prop. 2.) — 2 + 4 hyp. log. 2. 



Cor. Hence, as 1 + i + ^ + f^ -f &c. = 2, we have 1 + ^ + ^ + &c. 



= 2 hyp. log. 2, and i + ^ + ^^ + &c. = 2-2 hyp. log. 2. 



Prop. 10. To find the sum of- the infinite series 1 — 4 + 75 — 55 + &c. 

 temo- <Ae reciprocals of the figurative numbers of the 4th order, having the 

 signs alternately + ^^^d, — . 



If we write 2, — 3, 4, — 5, &c. for x, the general term will be , _ ■ = -, 



_j — . _j_ _ _j_ &c.; hence the sum required = 6/; + 6ci + 6/"+ &c. = (by prop. 



3) - 7i+ 12hyp.log. 2. 



Cor. Because the sum of 1 + 4 + 77, + ^ + ^c- = 2 ' '^herefore l + — + 



+ ^+&c. = -3 + 6 hyp. log. 2, and i + f^ + 1 -f &c. = 4^-6 hyp. 



log. 2. 



2^ 31 42 



Prop. 1 1. To find the sum of — — ^ + — — — + -^ — — + &c. ad infinitum. 



The general term, by writing 2, 3, 4, &c. for x, is _ .^n '^ . rr = -, + 



- + -6 + &c.; hence the sum = a + 2c + 3e + &c. = .884966993407. 



Prop. 12. To find the sum of 7—^7^ + JTp— + J7^— 3 + &c. at/ in- 

 finitum. 



Here the general term, by writing 2, 3, 4, &c. for a:, is r — i = 



1 2,4,6 9 12 , 16 5 ^, ^, 



i* ~ 7' + ^ + x» — ^ ~ I^ + ^"^ ~ ^^•' consequently the sum = e — 2p 



+ 4g — 6h -1- 91 — 12k + &C. = .010370898482. 



Prop. 13. To find the sum of \x — -i^B + J-c — &c. ud infinitum. 



The hyp. log. 2 = 1 - ^ + i - ;- + } - &c. = 1 + i + i + &c. - i 

 ~" i ~\ ~ ^^''' ^^^^^ ^ ^ ^y^- '°S- ^5 or hyp. log. 4, =?. + |+|^- 



