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

MULTIPLE POINTS.

11

tangents at a double point, both of which have a contact of the
second order with the curve. The preceding definition is consequently wanting in accuracy; and we shall therefore define a
tangent at any point of a curve as the line of closest possible
contact with the curve at that point.
We shall now resume the consideration of multiple points.
18. If a curve be referred to a point on itself as origin, the
linear term equated to zero is the equation of the tangent at the
origin.
The general equation of a curve of the nth degree when
expressed in polar coordinates may be written
A +(Bcos + Csin ) r + (D cos2 0 Esin20 +F sin20) r2
+ U+... Un = 0........................ (4).
When the origin lies on the curve A = 0, and one value of r is
zero; if, however, 0 be determined so that
B cos 0 + Csin = 0..................... (5),
two values of r will be zero, and the line
Bx + Cy= O
is the tangent to the curve at the origin.
19. If the origin be a double point the term of lowest dimensions
is the quadratic term, and this term equated to zero is the equation,
of the tangents at the double point.
If B = C = 0 two values of r will be zero whatever the value of
0 may be; and every line passing through the origin will have a
contact of the first order with the curve. If, however, 0 be determined so that
D cos2 0+ Esin 20 + Fsin2 0 =0...............(6),
three values of r will be zero, and the two lines whose inclinations
to the axis of x are determined by (6) will have a contact of the
second order with the curve. The origin is therefore a double
point, and (6) gives the directions of the tangents at the origin.
Their equation is
Dx2 + 2Exy + Fy2 = 0..................... (7).
It appears from (7) that the two tangents at the double point
will be real, coincident or imaginary according as E2 > or = or
< DF, in which three respective cases the origin will be a node, a