### On the cardioids fulfilling certain assigned conditions

Catholic University of America, 1917

Introduction

The tricuspidal, bicircular quartic of the third class defined by the

Cartesian equation (x^{2}+y^{2}+ax)^{2}=a^{2}(x^{2}+y^{2})

polar equation ρ=a(1-cos θ)

and commonly known as the Cardioid, has for many years been the object of mathematical investigation. It has lately been studied by Raymond Clare Archibald in his Inaugural-Dissertation "The Cardioide and Some of Its Related Curves" (Strassburg, 1900), which work contains an historical sketch of the curve and a presentation of results prior to the year of its publication. Since then, the only work on the subject of considerable length is Professor Archibald's paper, "The Cardioid and Tricuspid: Quartics with Three Cusps.". Besides this there have appeared a few detached problems and contributions in periodicals treating the curve from either a metric or a projective standpoint.

The chief characteristic of former research along this line seems to be the examination of the cardioid as a fixed curve and the consideration of its properties as such. The present investigation starts from a different point of view, which we may outline as follows:

in general, a curve of the fourth degree is capable of satisfying 14 conditions. The cardioid, however, having 3 cusps, two of which are at the fixed (circular) points I and J, can be subjected to only 4 conditions. If, then, 3 conditions be imposed, there are ∞^{1} curves satisfying them; therefore, the special elements (cusp, focus, double tangent) describe definite loci. If 4 conditions are given, there are a finite number of curves satisfying them. It is our purpose to obtain the loci generated in the first case; and, in the second, to determine the number and (where possible) the reality of cardioids for various kinds of assigned conditions.