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

42

TANGENTIAL COORDINATES.

The tangential equation of the evolute of an ellipse is
a2/ + b2/n2 = (a2 b2)2,
and therefore the evolute is a curve of the fourth class, and the
line at infinity is a double tangent which touches the curve at
two imaginary points. Hence the orthoptic locus is a sextic curve,
whose equation can be shown to be
(a2 + b2) (x2 + y2) (a2y2 + b2X2)2 = (a2 _ b2)2 (a2y2 - b22)2.
The Circular Points at Infinity.
69. It is proved in treatises on Trilinear Coordinates' that
the equation of every circle can be expressed in the form
S + (l m/3 + nry) I = 0,
where S is any given circle, and I is the line at infinity. The
constants (1, m, n) determine the position of the circle and its
radius; whilst the form of this equation shows that all circles pass
through the points of intersection of a given circle with the line at
infinity. These two points, which are imaginary, are called the
circular points at infinity and are usually denoted by the letters I
and J.
If S= 0 be the equation of the circle circumscribing the
triangle of reference, the circular points are the intersections of
S= 0, I=; that is of
/3y sin A + ya sin B + a/3 sin C= 0,
a sin A + 3 sin B + y sin C = 0.
Solving these equations, we obtain
a =-76LB, /=- -76LA...............(27),
which are the trilinear coordinates of the circular points at
infinity.
70. To find the Cartesian equations of the lines joining any
point with the circular points at infinity.
Let y = mx be the equation of any line joining the origin with
one of the circular points. The points of intersection of this line
with the circle x2 + y2 = a2 are given by the equation
l2 + 1 = a2/ 2........................(28).
1 Ferrers' Trilinear Coordinates, p. 87.