•28'2 PHILOSOPHICAL TRANSACTIONS. [aNNO 1758. 



^ 



(a— /a;t — l)", and A + V^AA— 1 = (« + ^aa— l)"": which values being 

 substituted above, we thence get 



— AXQog. (1— ^a:)+log.(l— ra7)]-|-V^AA— 1 X [log. (J — 92-) — log. (l— ra:)] ; 

 of which the former part (which, exclusive of the factor a, is hereafter denoted 

 by m) is manifestly equal to— a X log. (l^qx) x (l—rx) by the nature of lo- 

 garithms = — A X log. \_\ — {g-{-r)x-\- qrx^^ = — - a X log. {l — 2oix-{- xx) 

 by substituting the values of q and r : which is now entirely free from imaginary 

 quantities. But in order to exterminate them out of the latter part also, put 2/ = 



log. (1 — 957)— log. (1 —rx;) then will y = ~^- + '*'' — — 9—rxi 



1 — gx 1 — rx 1 j+rXX+XX 



= — ;; — ;; ; — = , — - — ; — : — *, where — expresses the 



I 2etX + XX 1 2etx + XX 1 — ^HX + XX ^ 



fluxion of a circular arch (n) whose radius is 1, and sine = Zl^iSf . con- 



sequently 2/ will be = — 2 j^ — 1 X n: which, multiplied by V^aa — I, or its 

 equal / — 1 X ^ J — aa, gives 2 \/l — aa X n ; and this value being added to 

 that of the former part, found above, and the whole being divided by n, we 



thence obtain ~^^ "^ — ■-",XN,or - X (— co-s. q X m -f sin a X 2 n) for 



that part of the value sought depending on the two terms affected with q and r. 

 Whence the sum of any other two corresponding terms will be had, by barely 

 substituting one letter or value for another : so that, 



■—log. (1—^) . - 



— co-s. a X M -|- sin. q X 2 n 

 7 X < — co-s. q' X m' -|- sin. a' X 2 n' 



— co-s. q" X m'^ + sin. q" X 2 n" 

 .— &c. H- &c. 



will truly express the sum of the series proposed to be determined ; m, m', u", 

 &c. being the hyperbolical logarithms of 1 — 2 ax X ix, 1 — 2^^ -f- ^^) 1 — 2 yj; 

 4- x^, &c. n, n , n", &c. the arcs whose sines are 



:, &c. and a, q', ql", &c. the measures 



Vl — <«* x'^l — -S/3 x-Zl — yy 



\/i — 2*1 + XX Vl — 23x + XX Vi — 2yx+xx 



tion, bat what has been a dvan ced in respect to the roots of the equation z« — 1 = 0, will appear 



manifest. For if j ± v/jx— 1 be put = z, then will z" = x± Vxx— 0" = x ± ^xx — 1 : 



where assuming x = 1 = co-s. = co-8. 360° = co-s. 2 x 360° = co-». 3 X 360°, &c. th« 



equation will become z« = 1, or zn — 1 = Oj and the different values of x, in the expression 



, . r ^. 360° 2x360* . 



(j ± V XX — 1 ) for the root z, will consequently be the co-sines ot the arcs - , —^, — , Sec. 



these arcs being the corresponding subraultiples of those above, answering to the co sine x (= 1.) 



In the same manner, if x be taken =-1 = co-s. 180° = co-s. 3 X I80« = co-s. 5 x 180^ 



tec then will z" =— 1 , or z" -|- 1 = 0; and the values of x will, in this case, be the co-sines of 



180° ^ 180° , 180° „ ^ . 



, 3 X •, 5 X , &c. — Ong, 



n n n 



