ON THE SYSTEM OF CONICS WHICH PASS THROUGH THE 
SAME FOUR POINTS. 
[From the Quarterly Mathematical Journal, vol. II. (1858), pp. 
I consider the system of conics passing through the same four points; these 
points may be real or imaginary, but it is assumed that there is a real system of 
conics, this will in fact be the case if two conics of the system are real. The four 
points are therefore given as the points of intersection of two real conics, and it will 
be proper to assume in the first instance that the conics intersect in four separate 
and distinct points, none of them at infinity. The four points may be all real, or 
two real and two imaginary, or all imaginary. 
First, if the points are all real, we have here two cases, viz. each of the points 
may lie outside of the triangle formed by the other three, or as this may be expressed, 
the points may form a convex quadrangle; or else one of the points may be inside 
the triangle formed by the other three, or as this may be expressed, the points may 
form a triangle and interior point. In each case the pairs of lines joining the points, 
two and two together, will be conics (degenerate hyperbolas) forming part of the system 
of conics. Consider the two cases separately.