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

CHAPTER III.
TANGENTIAL COORDINATES.
52. IN the Cartesian or the trilinear system of coordinates, a
curve is defined as the locus of a point which moves in a prescribed
manner. This condition leads to a functional relation between the
coordinates of the moving pbint, which is called the equation of
the curve.
In the tangential system, a curve is defined as the envelope of
a line which moves in a prescribed manner. Since the position of
any straight line is completely determined by means of two independent quantities, the condition that the line should move in the
prescribed manner involves a relation between these quantities,
which is called the tangential equation of the curve.
The system of tangential coordinates which we shall now
explain was invented by the late Dr Booth1 and is sometimes
called the Boothian system. Let ~ and X be the reciprocals of
the intercepts which a straight line cuts off from the axes; then
the equation of the line is
+ y =...........................(1),
and if this line envelopes a curve, a relation must exist between t
and X of the form
F(6 ) = 0......................... (2),
which is the tangential equation of the curve.
53. To find the tangential equation of a curve whose Cartesian
equation is given.
Let the Cartesian equation of the curve be

1 A Treatise on some New Geometrical Methods.