342] ON THE CONICS WHICH PASS THROUGH THREE GIVEN POINTS &C. 259 
The inverse coefficients are 
(“A ~f> gh, hf fg), 
and hence the condition in order that the conic may touch the line ax + (3y + yz — 0 is 
(A g\ h\ - gh, - hf, -fg\d, (3, yf = 0, 
or, what is the same thing, 
V (a/) + V (/3g) + V (yh) = 0. 
Similarly the condition in order that the conic may touch the line lx + my + nz = 0 is 
V (If) + V (mg) + V (nh) = 0. Hence if the conic touch the two lines, we have 
V(/) : V (g) ' V (h) = V (fin) — V (7to) : f (yl) — f (an) : f (am) — f (/31), 
or, what is the same thing, 
f ■ g ■ h = /3)i + ym — 2 V (¡3ymn) : yl + an — 2 V (yanl) : am + ¡31 — 2 f (a/3lm), 
which, since the radicals must be so taken that the product may be =a/3ylmn, gives 
in all four conics: and these will be all real if the signs of (l, m, n) are the same 
with, or opposite to those of (a, /3, y) respectively; which proves the theorem. 
In particular since infinity is a line meeting the three sides produced; if the 
given line meet the three sides produced, the system will contain four real parabolas 
but, if the given line meets two sides and a side produced, there is not any real 
parabola. In the latter case, as is obvious geometrically, the conics of the system are 
all hyperbolas. 
Any side of the triangle, and the line joining the opposite vertex with the point 
of intersection of the side and given line, form a pair of lines passing through the 
three points and meeting on the given line; such pair of lines is a conic of the 
system; and we have thus three pairs of lines, each pair a conic of the system. 
We may by what precedes form some idea of the nature of the system of conics 
which pass through the three given points and touch the given line. In fact writing 
at the point of contact the letters H, P, E, L according as the conic is a hyperbola, 
parabola, ellipse, or pair of lines, then if the given line cut the three sides produced, 
we have as in fig. 1. 
Fig. 1. 
33—2