Since c' meets AB at { (n —p — k) + (n — q — k) + k } points, it
is a (2n—p — q — k)-ic. Hence :

If c is an n-ic with a k-ple point at G, a p-ple point at A,
and a q-ple point at B, then c' is a (2n—p — q — k)-ic with an
(n — p — q)-ple point at G, an (n-p-k)-ple point at A, and
an (n — q — k)-ple point at B.
For instance, taking n = 2, p = q = 1, k = 0, we see that
the transform of a conic through A and B is a conic through
A and B.
Ex. 1. Properties of any quintic with three or more double points can
be deduced from those of a quartic with the same deficiency.
[Take three double points as A, B, C of § 1. The properties so derived
are not usually of much elegance or importance. Our knowledge of the
theory of quintic curves (and still more of sextic) is very limited.]

Ex. 2. If two curves (or branches of the same curve) have r-point
contact at a point on CA, the transformed curves (or branches) have
(r+ l)-point contact at A.
§ 2. Inversion.
If in § 1 we project A, B into the circular points, 2 becomes
a circle with centre C, and P, P' are inverse points with
respect to this circle. The loci c, c' of P, P' are now inverses
of each other with respect to the circle. Hence inversion is
a particular case of quadratic transformation, and quadratic
transformation is the generalization by projection of inversion.
