The object of this paper is to show that the leading properties of the Cartesian oval, as developed in such texts at Williamson's "Differential Calculus," Salmon's "Higher Plane Curves" or Loria's "Spezielle Algebraische und Transcendente Ebene Kurven," may be readily deduced from the Weierstrass elliptic functions p and σ, and that they give a geometric interpretation to the standard formulae of these functions.
Greenhill has used these functions in connection with the Cartesian oval (Proc. London Math. Soc., t. XVII, 1886, and also his treatise on "Elliptic Functions," Arts. 236, 248 and 249) and has deduced the relations between the focal radii, though we have give a more symmetric form to these relations by the introduction of the triple focus.
Professor Frank Morley published a note on the subject some years ago in the Haverford Studies. (See also Harkness and Morley, "Treatise on the Theory of Functions," p.336.)
When the u-plane is mapped to the x-plane by means of the equation
where Q is a quartic in x, to lines in the u-plane parallel to the real or imaginary axis, correspond in the x-plane bicircular quartics whose real foci are the zeros of Q, provided these zeros are concylic or anticyclic. (Greenhill, Camb. Phil. Proc., t. IV; Franklin, American Journal of Mathematics, Vol. XI, 3 and Vol. XII, 4.)
The problem of reducing this form to the Weierstrass form which we use, where one focus is at infinity, is analogous to the reduction of the bicircular quartic to the Cartesian. (Salmon, "Higher Plane Curves," Art. 280.)