^ + ^ + ^=0. 
[191 
192] 
143 
the 
nsion 
and 
192. 
ON THE AREA OF THE CONIC SECTION REPRESENTED BY 
THE GENERAL TRILINEAR EQUATION OF THE SECOND 
DEGREE. 
[From the Quarterly Mathematical Journal, vol. n. (1858), pp. 248—253.] 
[The original title was “Direct Investigation of the Question discussed in the 
Foregoing Paper,” viz. a Paper with the present title by N. M. Ferrers (now 
Dr Ferrers), pp. 247—248. The area S of the conic section represented by the 
general equation (A, B, C, A', B', G'\x, y, zf = 1, where the coordinates are con 
nected by the equation x + y + z = 1, was by considerations founded on the form of 
the function found to be 
27r (AA' 2 + BB' 2 + GC' 2 - ABG - 2A'B'C') A 
^ ~ {A' 2 -BG+ B' 2 — GA + G' 2 - AB + 2 (B'G' — AA') + 2 (G'A' - BB') + 2 (A'B' - GG')\ h ’ 
where A is the area of the fundamental triangle: and it was remarked that a 
similar method might be applied to determine the area of the conic section when 
it is defined by the distances of its several tangents from three given points.] 
The position of a point P being determined as in the foregoing paper, let 
a, /3, y denote in like manner the coordinates of a point 0, we have 
a + /3 + y = 1, 
and consequently if f, y, £ are the relative coordinates x — cc, y — ¡3, z — y, we have