An elementary treatise on cubic and quartic curves, by A. B. Basset.

26

THEORY OF CURVES.

Let ABC be the triangle of reference, and let AB'C' be a
subsidiary triangle of reference such that the base B'C' cuts the
lines AB, AC in B' and C'. Let (a, /3, y) and (a', /, ry) be the
trilinear coordinates of a point referred to the two triangles, where
a'= 0 is the equation of B'C' referred to ABC, and consequently a'
is a linear function of a, 3, 7.
The equation of a curve having any proposed singularity at B'
can be at once written down whenever the nature of the singularity
is known. If, however, B'C' be supposed to move off to infinity,
B'C' will become the line at infinity, and its equation referred to
ABC will be I = 0, where
I = aa + b/3 + cy;
consequently the trilinear equation of a curve having any proposed
singularity at infinity upon the line AB may be obtained by first
writing down the trilinear equation of a curve having the proposed
singularity at B, and then changing a into I.
The general equation of a curve having a double point at
B is
/3n-2 2 + /n-3U3 +... = 0............... (32),
where u, is a binary quantic in a and 7. Hence the general
equation of a curve having a double point at infinity on the line
AB is of the same form as (32), where u, is a binary quantic in I
and y.
48. To find the equation of a curve having a double point at
infinity on the axis of x.
Let the triangle of reference have a right angle at A, and let
AB and AC be the axes of x and y. Then the trilinear equation
of a curve having a double point at B is given by (32). Let
u2 = Xa2 + 2/ary + ry2;
then when B moves off to infinity, we must write
a=I, 3=x, =y.....................(33),
where I is constant, whence (32) becomes
in-2 (X-2 + 2I y + vy2) + Xn-3 U3 +... Un = 0....(34),
where U,, is a polynomial of the nth degree in y. Equation (34)
is the general equation in Cartesian coordinates of a curve which