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Descartes' and Hessian Foliums.(geometric exercises)
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Abstract.--The conic polars of [x.sup.3] + [y.sup.3] = 3axy are called the Hessian Foliums. The set of areas of the loops of this set becomes a convergent infinite series. Also the set of lengths of these loops gives another convergent series (Bagchi 1939).
1. THE FOLIUM
The equation of the DesCartes' folium is
[x.sup.3] + [y.sup.3] = 3axy. (1)
For simplicity let a = 1. Then one obtains a set of parometric equations for (1) by the following set of equations
{[x.sup.3] + [y.sup.3] = 3xy
y = tx. (2)
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