52 ON THE CONICS WHICH TOUCH THREE GIVEN LINES, &C. [393
the condition in order that the conic may pass through the given point is a + b + c = 0,
and we thus find for the curve, which is the locus of the point S, the equation
1 1
VO) + V(2/) + VO)~ ’
or, what is the same thing,
V (yz) + V (zx) + V (xy) = 0,
the rationalised form of which is
y 2 z 2 + z 2 x 2 + x 2 y 2 — 2xyz (x + y + z) = 0.
This is a quartic curve with three cusps, viz. each angle of the triangle is a cusp;
and by considering for example the cusp (y = 0, z = 0) and writing the equation under
the form
a?(y- zf - 2x (;yz 2 + y-z) + y 2 z 2 = 0,
we see that the tangent at the cusp in question is the line y — z — 0 ; that is, the
tangents at the three cusps are the lines joining these points respectively with the
given point (1, 1, 1). Each cuspidal tangent meets the curve in the cusp counting as
three points and in a fourth point of intersection, the coordinates whereof in the case
of the tangent y — z = 0, are at once found to be x : y : z = 1 : 4 : 4, or say this is
the point (1, 4, 4) ; the point on the tangent z — x — 0 is of course (4, 1, 4), and that
on the tangent x — y = 0 is (4, 4, 1). To find the tangents at these points respectively,
I remark that the general equation of the tangent is
that is
X Y Z
-| + ~S +
X* y*
or for the point (1, 4, 4) the equation of the tangent is 8X+Y+Z = 0, or say
8x + y + z = 0 ; that is, the tangent passes through the point x = 0, x + y + z = 0, being
the point of intersection of the line x = 0 with the line x + y + z = 0, which is the
harmonic of the given point (1, 1, 1) in regard to the triangle; the tangents at the
points (1, 4, 4), (4, 1, 4), (4, 4, 1) respectively pass through the points of intersection
of the harmonic line x + y + z — 0 with the three given lines respectively.
In the case where the given point lies within the triangle, the curve the locus
of S lies wholly within the triangle, and is of the form shown in fig. 3 in the plate
opposite ; it is clear that in this case the conics of the system are all of them ellipses;
there are however three limiting forms, viz. the line joining the given point with any
angle of the triangle, such line being regarded as a twofold line or pair of coincident
lines, is a conic of the system. The discussion of the two cases in which the given
point lies outside the triangle, viz. in the infinite space bounded by two sides produced,
or in the infinite space bounded by a side and two sides produced, may be effected
without much difficulty.
