392] 
43 
392. 
ON THE CONICS WHICH PASS THROUGH TWO GIVEN POINTS 
AND TOUCH TWO GIVEN LINES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), 
pp. 211—219.] 
Let x = 0, y = 0 be the equations of the given lines; z = 0 the equation of the 
line joining the given points. We may, to fix the ideas, imagine the implicit constants 
so determined that x + y 4 z=0 shall be the equation of the line infinity. 
Take x — my = 0, x — ny = 0 as the equations of the lines which by their inter 
section with z = 0 determine the given points. The equation of the conic is 
{V (m) + V (0} V (xy) = x+yf (mn) + 7z, 
or, what is the same thing, 
(x — my) (x — ny) + 2 [x+y f (mn)} 7z + 7 2 z 2 = 0, 
so that there are two distinct series of conics according as V (mn) is taken with the 
positive or the negative sign. 
The equation of the chord of contact is 
x + y V (mn) + yz = 0, 
which meets z-0 in the point [x + y f(mn) = 0, z = 0} that is in one of the centres of 
the involution formed by the lines (x — 0, y = 0), (x — my = 0, x — ny = 0). It is to be 
observed that the conic is only real when mn is positive, that is (the lines and points 
being each real) the two points must be situate in the same region or in opposite 
regions of the four regions formed by the two lines: there are however other real 
cases; e.g. if the lines x = Q, y = 0 are real, but the quantities m, n are conjugate 
imaginaries; included in this we have the circles which touch two real lines. 
6—2