VOL. LXVIir.] PHILOSOPHICAL TRANSACTIONS. 36l 



the steps which have been traced out by these imaginary operations. Let abe be 

 an equilateral hyperbola (fig. 5) of which the centre is f, and the transverse axis 

 AF = 1 ; let ABF, ACF, ADP, &c. be any sectors in arithmetical progression, and 

 let TO be their number; it is required to find the sum of all the ordinates eg, ch, 

 DK, &c. belonging to those sectors. Let the sector afb = -^a, and the sector 

 BFC, which is the common difference of the sectors, = J^x: then bg, or ord. a 

 —-•ilzf — J and CH, or ord. a + x = ~ , by art. 5. Therefore the series 



of ordinates, that is, bg + ch + dk. -)- (m) = — x 



(m) - ^' X (1 + c-^ + C-2X + (^) = T 



J _ C-J»v c^ — C" -'t '"" — c'>-''-\-C<' + >"x—x — c^''-\-C — •'—'">' -\-c-i' — 



ord. a - ordJ a+ m.) - ord. (. - .) + ord. (a + m.-. )_ When « = :i, Ord. .r + ord. 2^ 

 2 X ( 1 — abs. x) 



, . ord. X — ord. (n* + 1) x .r + ord. mx 



+ 0'-d-3^ + (^) = 2x(l-abs..) • 



In like manner it is proved that the sum of the abscissae, that is, fg + fh 

 . . abs. a — abs. (a + w?i) — abs. (a — x) + abs. (a + mx — x) , , 



+ PK + (m) = 2x(i-abs.x) ^ ' ^"d ^hen 



, abs. mx — abs. (m + 1 ) x x 

 a = .r, this expression becomes 2x (i-abs.x) -^• 



10. The coincidence of the theorems deduced in the last two articles is obvious 

 at first sight, and if the metiiods by which they have been obtained be compared, 

 it will appear, that the imaginary operations in the one case were of no use but 

 as they adumbrated the real demonstration which took place in the other. This 

 will be rendered more evident by considering that the resolution of the series of 

 hyperbolic ordinates, into two others of continual proportionals, can be exhibited 

 geometrically. For, from the points a, b, c, and d, let am, bn, co, dp, be 

 drawn at right angles to the assymptote fp; let gb produced meet fp in q, and 

 let BR be perpendicular to the conjugate axis fr. Then, because the triangles 

 frs fma, are equiangular, af : fm :: fs : fr; hence fr = -^ X ps = — x 



(fn _ nb). For the same reason ch = — X (fo — oc) and dk = — x (fp 



— pdV Therefore, bg + ch + dk = — - X (fn -f fo + fp) — — x (bn + 



/ FA \ A. 



CO + dp) ; now, fn, fo, fp, are continual proportionals, and so also are bn, fo, 

 FP because the sectors fbc, fcd, are equal. But in the circle no such resolu- 

 tion of the proposed series of sines can take place, that series being subject to 

 alternate increase and diminution; on which account it is, that imaginary cha- 

 racters enter into the exponential value of the sine. Those characters are there- 

 fore so far from compensating each other in the present case, as they ought to 

 do, on the supposition of Bernoulli and Maclaurin, that they manifestly serve as 

 marks of impossibility. There remains, of consequence, the affinity between 

 VOL. XIV. 3 A 



