262
ON THE CONICS WHICH PASS THROUGH
[342
The equation of the curve of centres was given in the late Mr Hearn’s “ Researches
on Curves of the Second Order, &c. London, 1846,” viz. if x — 0, y = 0, z — 0 be the
equations of the sides of the triangle formed by the given points; x + y + z = 0 the
equation of the line infinity, and ax + /3y + yz = 0 the equation of the given line, then
the equation of the curve of centres is
V [ax (- x + y + z)) + V {/3y (x - y + z)) + V {yz {pc + y-z)} = 0,
or more generally if x + y + z = 0 be the equation of an assumed line, then this equation
is that of the locus of the pole of the assumed line in regard to the conics passing
through the given points and touching the given line, see my paper “Note on a Family
of Curves of the Fourth Order,” Cambridge and Dublin Mathematical Journal, t. v. (1850),
pp. 148—152, [85], where I have noticed the above mentioned property, that the conic
through the points of contact of the tangents through the nodes breaks up into a
pair of lines. It is I think worth while to show how the equation is obtained. The
equation of a conic through the given points and touching the given line is
(0, 0, 0, f g, h$se, y, zf = 0
with the condition V («/*) + V (/%) + V (y^) = 0, and this being so, the coordinates of
the pole in relation thereto, of the assumed line x + y + z = 0, are
x : y : z= (~f + g + h)f
■ ( f~9 + h)g
: ( f+g-h)h.
We have thence
— x + y + z proportional to — (— f+ g + h)f
+ ( f-g+ h )g
+ ( f+9-h)h,
that is, to f 2 —(g~h) 2 , which is = (/— g + h) (f+g — h),
and combining with this the equation
ax proportional to
we obtain
that is
ax (— x + y 4- z) proportional to
(~f+g + h)fa,
*/>
ax (— x + y + z) : ¡3y (x — y+z) : yz (x +y — z) = af : f3g : yli,
so that from the equation V {af) + J (/3g) f \/ (y/i) = 0, we have at once the
equation
V (~ « + y + z)) + V {/3y (x - y + z)} + V [yz (x + y - z)} = 0.
The rationalised form is
foregoing
(1, 1, 1, — 1, -1, - l$a x(-x + y + z), ßy(x-y + z), yz (x + y- z)) 2 = 0,
