t 287 ] 
Fig. 2. In the ellipfis aed, the lemi-tranfverfe axis 
cdis=^?+^2; the femi-conjugate ca=<5; the abfciffa 
= srid the ordinate be=^x -K 
a ' a\ 
Fig. 3. In the elliplis aefd, the femi-tranfverfe axis 
cd is -\,d" + + 7 ^ ; the femi-conjugate 
the tangents ep, fq, intercepted by perpendiculars (cp, cq) 
drawn thereto from the center c, each -a^x and 
the abfciffa (c b' or cb'^) on cd,correfponding to the point 
e or f, of the curve is determined by the expreflioii 
+ + — Z + Z* + — z| 
cd. 
The quadrantal arc ad (fig. 2.) is denoted by E", and 
the quadrantal arc ad (fig. 3.) is denoted by f. l the 
limit of DP- AD (fig. I.) is = 2 F-E. 
From what is now done, I might proceed to deduce 
many other new theorems, for the computation of 
fluents; but I fhall, at prefent, decline that bufinefs: 
and, after giving a remarkable example of the ufe of 
theorem 4. in computing the defcent of a heavy body 
in a circular arc, conclude this paper with a few obfer- 
vations relative to the contents of the preceding articles. 
5, Let Ipqn (fig. 4.) be a femi-circle perpendicular 
to the horizon, whofe higheft point is 1, loweft n, and 
center m. Let ps, q t, 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 and the diftance 
VoL. LXV. R r St by 
