If thirteen basal points coincide, the system of quartics then consists of the three fixed lines joining the multiple point to the other three, together with a pencil of lines through the multiple point. If fourteen points coincide, two lines are fixed and these with any two lines of the pencil form a quartic of the system. If fifteen points coincide, only one line is fixed and each quartic consists of this line and any other three of the pencil. When all sixteen points coincide, any four lines through it form a quartic of the system.
In this paper cubic and quartic curves only are considered. I expect in a future paper to extend the methods herein developed to curves of still higher degrees. Many of the present results can be generalized and stated for a unicursal curve of the nth degree. I have purposely omitted all consideration of focal properties of these curves. There are also many special forms of interest which do not properly belong to a general treatment of the subject.
The theorem concerning the three points on a conic A, B, and C, whose osculating circles pass through a fourth point O on the conic, is due to Steiner. From the properties of the harmonic polars of the points of inflection on a nodal cubic we may infer many other theorems concerning the points A, B, and C on a conic. Let the cubic be projected into a circular cubic and then inverted from the node. Its points of inflection A₁, B₁, C₁ invert into the points A, B, and C. The harmonic polar of A₁ inverts into the common chord O P of the circles osculating the conic at B and C; and similarly for the other harmonic polars.
The pencil O {A B P C} is harmonic. Any circle through A and O meets the conic in S and T so that the pencil O {A S P T} is harmonic. The two circles through O and tangent to the conic at S and T intersect on O P. If two circles be drawn through O and A intersecting the conic one in S and T and the other in U and V, the circles O S U and O T V intersect on O P; so also the circles O S V and O T U. But one circle can be drawn through O and A and tangent to the conic; its point of contact is on O P. Let l, m, and n be three points on the conic on a circle through O. Draw the circles O A l, O A m, and O A n intersecting the conic again in l₁, m₁, n₁; l₁, m₁, n₁, are also on a circle through O, and the circles through l, m, n and l₁, m₁, n₁ intersect on O P.
From the fundamental property of the Cissoid of Diocles we can obtain by inversion an interesting theorem concerning the parabola. In the figure of the Cissoid given in Salmon’s H. P. C. Art. 214, A M₁ = M R, whence A M₁ = A R - A M; or A R = A M + A M₁. Inverting from the cusp and representing the inverse points by the same letters, we have for the parabola
| 1 | 1 | 1 | ||
| ——— | = | ——— | + | ——— . |
| A R | A M | A M₁ |
This result is interpreted as follows:—draw the circle of curvature at the vertex of a parabola; this circle is tangent to the ordinate B D which is equal to the abscissa A D; draw a line through A cutting the circle in R, the ordinate B D in M, and the parabola in M₁; then
| 1 | 1 | 1 | ||
| ——— | = | ——— | + | ——— . |
| A R | A M | A M₁ |