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VOL. LXV.] PHILOSOPHICAL TRANSACTIONS. 649 



These theorems still refer to fig. 11, 12, 13; but now the values of the several 

 lines in them (being not as before) are as here specified, viz. Fig. 11, in the 

 hyperbola ad, the semi-transverse axis ac is now = a; the semi-conjugate := b; 

 the perpendicular cp, from the centre c on the tangent dp, is = \/ az, the said 



tangent dp = \/- X (i^ + 2^z — z'^); and the abscissa CB (corresponding to the 



ordinate bd) is =i a^/ - X \/ ' . .. • 



' r a' + b^ 



Fig. 2. In the ellipses aed, the semi-transverse axis cd is = V'(a* + b'^); the 

 semi-conjugate ca = b; the abscissa cb = 

 a/ "—^ — X */ (a — z) ; and the ordinate he=b t/- . 



Fig. 13. In the ellipsis aefd, the semi-transverse axis cd is = -j- v^(a* X b'^) + 

 \a\ the semi-conjugate ca ^ -^ v/(o* -|- b'^) — t«; the tangents ep, fq, inter- 

 cepted by perpendiculars (cp, cq) drawn to them from the centre c, each = -/ a 

 X (a — z) ; and the abscissa (cb' or cb") on cd, corresponding to the point e or f, 



of the curve is determined by the expression ^— /,») ^ ^' 



. The quadrantal arc ad (fig. 1 2) is denoted by e ; and the quadrantal arc ad (fig. 

 13) is denoted by f; l the limit of dp — ad (fig. 1 1) is = 2f — e. 



From what is now done, I might proceed to deduce many other new theorems, 

 for the computation of fluents ; but I shall at present decline that business : and, 

 after giving a remarkable example of the use of theorem 4, in computing the 

 descent of a heavy body in a circular arc, conclude this paper with a few obser- 

 vations relative to the contents of the preceding articles. 



5. Let Ipqn (fig. 14) be a semi-circle perpendicular to the horizon, whose 

 highest point is 1, lowest n, and centre m. Let ps, qt, parallel to the horizon, 

 meet the diameter Imn in s and t: and let the radius Im (or mn) be denoted by r, 

 the height ns by d; and the distance st by x. Then, putting h for (l6-iV feet) 

 the space a heavy body, descending freely from rest, falls through in one second 

 of time ; and supposing a pendulum, or other heavy body, descending by its 

 gravity from p, along the arc pqn, to have arrived at q; the fluxion of the time 



of descent will be = ,, , — ^- — -^, — " ,. ,, r^. The fluent of which, or the 



^yZrd — d?' — 1 (r — a) (d — i)] 



time of descent from p to q is, by theorem 4 of the preceding article, = 

 — T — T^rzrA\ >^ ^^ ~" ^^5 '* ('" •^hat theorem) being taken z=z\/ d, b = v^(2r — a), 

 cb (fig. 2) = v^ — X ■/ (<^ — ar), and ep, fq, (fig. 1 3) each =4/ {d — x). Hence 



2/* 



it appears, that the whole time of descent from p to n is = —7 — - ■ _ - x (a 



— f) ; when, in fig. 1 2 and 1 3, the semi-axes are taken according to the values of 

 a and /; just now specified. 



VOL. xiir. 4 O 



