VOL. L.] PHILOSOPHICAL TRANSACTIONS. 281 



same quantity must here be made = — 3 or — n. Whence/)" being = — i, 

 q" = — \, &c. the values of p, q, r, he. will in this case be the roots of the 

 equation z" + 1 = O. 



It may now be proper to set down an example or two of the use and applica- 

 tion of the general conclusions above derived. First then, supposing the series, 

 whose sum is given, to be 



^ '' "2 "^ 3 "■ 4 ' ot" "*" m + 1 "^ m+ 2 ••••"" ^IJT^ + m + 7i + l "^ 



&c. = — H. log. 1 — X (= s) ; let it be hence required to find the sum of the 



series ( \- —- 1 — -- &c.) arismg by taking every w' term, beginning 



with that whose exponent (m) is any integer less than n. Here the terms pre- 

 ceding — being transposed, and the whole equation divided by x*", we have 



^ + --n + ^+- + ^+-3' *"=• = - jr, X H. log. (I - x) - -4^—. In 

 which value, let px^ qx, rx, &c. be successively substituted for x (according to 

 prescript) neglecting entirely the terms — ~— , as having no effect at all in the 



result: whence we 2:et — - — — X loe. (1 — bx) — - — - X loo-. ( I— ax) ? — 



° CpxJ>" ° ^ ^ ^ CqxJ't ° ^ ^ ' (rx)"' 



X log. (1 — rx)j &c. Which multiplied by x"' (the quantity that before divided) 

 gives - ^„ X log. {i-px)-j^y< log. (1 - qx) - ;^ X log. (1 - rx), &c. 

 = n times the quantity required to be determined. 



But now to get rid of the imaginary quantities q, r, &c. by means of their 

 known values a -f- V «» — ■ I, a, — '^ ocx — 1, &c. it will be necessary to observe, that 

 as the product of any two corresponding ones (x -\- ^ xx — l) X {x—'^xx — 1) 

 is equal to unity, we may therefore write (a — ^ xx — l)" (rsr") instead of 

 its equal -„, and (x-\-^ xx — l)"* ( = 9*") instead of its equal — : by which 

 means the two terms, wherein these two quantities enter, will stand thus ; 



— {x— ^a* — l)"* X Log. {l—qx) — (a-f */ xx—l)"" X Log. (l— r^.) 



But if A be assumed to express the co-sine of an arch (a,) m times as great 

 as that ( ) whose co-sine is here denoted by a; then will a — "/aa— 1 =* 



• Because ■■ and — are known to express the fluxions of the circular arcs whose 



V] XX Vl — XX 



co-sines are x and x, it is evident, if those arcs be supposed in any constant ratio of 1 to n, that 



nx X* , , 1 '*•*' , nx X* 



— = ■ , and consequently that — -== (= ^ — = — = ) 



Vi — XX vl— XX v *x— 1 a' — 1 y.'^l—xx V — Ix'^l— XX 



X' 



Whence, by taking the fluents, n x log. (x -J- '^xx — l) or log. (x 4- '^xx — !)« = 



a/xx— 1 



log. (x -\- v^xx — 1 ;) and consequently (x+a'xx — \)n = x -|- v^xx — 1 : whence also seeing 

 X — Vxx — 1 is the reciprocal of x -|- v^.rx — 1, and x — v^xx — T of x + Vxx— 1, it isalso 

 evident that (x — ^kx — 0" = x — v^xx — 1. Hence, not only the truth of the above assump- 



VOL. XI. O O 



