[ 2^4 ] 
tangent dp and the arc ad will be equal to the fluent of 
7TI W 1' 
m + n\—t 
X 
2. It is well known, that in any ellipfis whofe femi- 
tranverfe axis is m, and femi-conjugate n', if a’ be the 
abfcifla, meafured from the center upon the tranfverfe 
axis, and z the arc between the conjugate axis and the 
— 
ordinate correfponding to .r, f x ^ will be = 
being = 
Hence, 
m 
7n 
being 
»7 + « 
t, it 
m~ 
m- 
:appears, that in the ellipfls aed (fig. 2.) whofe femi- 
tranfverfe axis cd is - m + n, femi-conjugate ca= 2 x ?nn^i 
and abfciflTa cb (correfponding to the ordinate be) 
/; the arc ae is equal to the fluent of /. 
3. In the ellipfis aefd (fig. 3.), the femi-tranfverfe 
;axis cd being = the femi-conjugate ca = n\ and 
the abfcilfa cb (correfponding to the ordinate be) = a"; 
if e p, the tangent at e, intercepted by a perpendicular 
(cp) drawn thereto from the center c, be denoted by t\ 
/gx X (as is well known) will be = if, g being as 
in the preceding article. 
Hence 
