This paper contains a summary of the work done during the last school
year by my class in Modern Geometry. Since many of the results were
suggested or entirely wrought out by class-room discussion, it becomes
practically impossible to assign to each member of the class his
separate portion. Many of the results were contributed by Messrs. M. E.
Rice, A. L. Candy, H. C. Riggs, and Miss Annie L. MacKinnon.
The reader who is not familiar with the method of Geometric Inversion
should read Townsend’s Modern Geometry, chapters IX and XXIV; or a
recent monograph entitled, “Das Princep der Reziproken Radien,” by C.
Wolff, of Erlangen.
When a conic is inverted from a point on the curve, the inverse curve
is a nodal, circular cubic.
This is shown analytically as follows: let the equation of the conic be
written
ax² + 2hxy + by² + 2gx + 2fy = 0;
which shows that the origin is a point on the curve. Substituting for
x y
x and y ————————— and ———————— ,
x² + y² (x² + y²)
we have as the equation of the inverse curve
ax² + 2hxy + by² + 2(gx + fy)(x² + y²) = 0.
The terms of the second degree show that the origin is a double point
on the cubic; and is a crunode, acnode, or cusp, according as the
conic is a hyperbola, ellipse, or parabola. The terms of the third
degree break up into three linear factors, viz: gx + fy, x + iy, and
x - iy, which are the equations of the three lines joining the origin
to the three points where the line at infinity cuts the cubic; thus
showing that the cubic passes through the imaginary circular points at
infinity.
Since the above transformation is rational, it follows that there
is a (1, 1) correspondence between the conic and the cubic. This
fact is also evident from the nature of the method of inversion. The
cubic has its maximum number of double points, viz: one; and hence is
unicursal. This unicursal circular cubic may be projected into the most
general form of unicursal cubic; the cuspidal variety, however, always
remaining cuspidal.
By applying the method of inversion to many of the well known theorems
of conics, new theorems are obtained for unicursal, circular cubics.
If one of these new theorems states a projective property, it may at
once by the method of projection be extended to all unicursal cubics.
Examples will be given below.
The following method of generating a unicursal cubic is often useful.
Given two projective pencils of rays having their vertices at A and
B; the locus of the intersection of corresponding rays is a conic
through A and B. Invert the whole system from A. The pencil through
A remains as a whole unchanged, while the pencil through B inverts
into a system of co-axial circles through A and B, and the generated
conic becomes a circular cubic through A and B, having a node at A.
Now project the whole figure and we have the following:—given a system
of conics through four fixed points and a pencil of rays projective
with it and having its vertex at one of the fixed points, the locus
of the intersection of corresponding elements of the two systems is
a unicursal cubic, having its node at the vertex of the pencil, and
passing through the three other fixed points.
Unicursal cubics are divisible into two distinct varieties, nodal and
cuspidal. The nodal variety is a curve of the fourth class and has
three points of inflection, one of which is always real. The cuspidal
variety is of the third class and has one point of inflection (Salmon,
H. P. C., Art. 147). Each of these varieties forms a group projective
within itself; that is to say, any nodal cubic may be projected into
every other possible nodal cubic, and the same is true with regard to
the cuspidal. But a nodal cubic can not be projected into a cuspidal
and vice versa.
In applying this method of investigation to the various forms of
unicursal cubics and quartics, only a limited number of theorems are
given in each case. It will be at once evident that many more theorems
might be added, but enough are given in each case to illustrate the
method and show the range of its application. It is not necessary to
work out all the details, as this paper is intended to be suggestive
rather than exhaustive.