296] 
505 
296. 
ON THE CONICS WHICH PASS THROUGH THE FOUR FOCI 
OF A GIVEN CONIC. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. v. (1862), 
pp. 275—280.] 
The foci of a conic are the points of intersection of the tangents through the 
circular points at infinity ; the pair of tangents through each of the circular points 
at infinity is a conic through the four foci; and we have thus two conics P = 0, Q = 0 
passing through the four foci; the equation of any other conic through the four foci 
is of course P + XQ = 0; and in particular if X be suitably determined this equation 
gives the axes of the conic. 
I was led to develope the solution, in seeking to obtain the elegant formulae 
given in Mr P. J. Hensley’s paper “Determination of the foci of the conic section 
expressed by trilinear coordinates,” Journal, t. v., pp. 177—183, (March, 1862). 
I take the coordinates to be proportionate to the perpendicular distances of the 
point from the sides of the fundamental triangle, each distance divided by the perpen 
dicular distance of the side from the opposite angle. This being so, the equation of 
the line infinity is 
x + y + z = 0, 
and, a, (3, 7 denoting the sides of the fundamental triangle, the equation of the circle 
circumscribed about the triangle is 
—1 1—- 
X y z 
The foregoing two equations determine the circular points at infinity; and if (x lt y lt zf 
are the coordinates, there is no difficulty in obtaining the system of values 
: V i '■ z x = -ct 
a (cos G —i sin G) : — /3 
a (cos B + i sin B) : /3 (cos A —i sin A ) 
/3 (cos G + i sin G ) 
7 (cos B + i sin B ) 
7 (cos A +ïsin A) 
C. IV. 
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