The tangents from to the inverse curve are the n tangents
at each counted twice (Ch. VII, § 5) and the tangents from
to the original curve. Hence m' = m + 2n.
The same result follows from consideration of the tangents
from a) (or a/), remembering that a line through co inverts into
a line through co'.
The result If = D is a particular case of the theorem that
two curves with a 1:1 correspondence have the same defi-
ciency (Ch. XXI, § 3).
We have proved earlier (Ch. V, § 4) that the inverses of the
foci of a curve are the foci of the inverse curve ; and that, if
is a focus of the curve, the inverse curve has cusps at
a> and co'.
Ex. 1. If the original curve has a fc-ple point at and a #-ple point
at each of a> and ta', we have
n' = 2n-2p-k, 6' = $(«-2p) (n-2p-l) + (n-p-k) (n-p-k-\) + h,

k = k ;
and the inverse curve has a (m-2p)-ple point at and (n-p-k)-j>\e
points at each of <» and <»'.
Ex. 2. What modification is required in the result of § 3, if the curve
touches (bo)' ?

* In fact any line x±iy = c meets the inverse curve at n finite points
only ; whereas the curve i9 of degree 2«.
166 INVEESE CURVE XI 3

Ex. 3. A curve of degree 2N with .ZV-ple points at <u and a' inverts
into a curve of the same type.
Ex. 4. Find the number of circles of curvature of a given curve which
pass through a given point 0. Discuss the case in which lies on the
curve.
