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
Draw the circle with centre at D and radius A D; any chord of the
parabola through the vertex is cut harmonically by the parabola, the
circle, and the double ordinate through D.
Footnotes:
[1] See note A.
[2] A few of the results of this section are due to the late Mr. H. B.
Hall.
[3] See note B.
Foreign Settlements in Kansas.