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

16

THEORY OF CURVES.

Polar Curves.
23. Before considering the theory of polar curves, it will be
convenient to explain the notation that will be employed, and also
to prove a preliminary proposition.
The letters a, 13, y will be employed to denote the trilinear
coordinates of a variable point; the letters (f, g, h) will denote
the coordinates of a fixed point in the plane of the curve; and the
letters &, r, ' the coordinates of a fixed point on a curve. Also
the letters A, A' will be used to denote the operators
d d d
A ==d-/+g0+hdr,y
d d d
a = a d- +/3 d+ ~Y
ot/ c^7 dh'
If F(x, y, z) be any ternary quantic of degree n, and if a +f,
+g, y + h be written for x, y, z, then
'APF A/n-p F ( h)n-.
p F (a, -, v )=( _ p)! F (f, g, h).
By Taylor's theorem,
F(a +f,3 +g, y+h)= F(a, /3, )+ +... F+ n....(15),
and
F(a +f, /3 + g, y + h) = F(f,g, h) + A'F'+ ++ F'...(6),
where F'=F(f, g, h). Since F is a homogeneous function it
follows that
F(a+f, +g, y + h)=F(a,, 7)+ F(f, g, h) +P,
where P consists of a series of products into which at least one of
the quantities a, /3, 7 enters into combination with at least one of
the quantitiesf, g, h.
Since A/2F does not contain a, 3, y it follows that
F(f, g, h)= I AnF(a, /3, 7)
n!

similarly

F (a, 3, 7) = 1 A'fF(f, g, h),
n.