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In an article from 1865, Arthur Cayley claims that given a plane algebraiccurve there exists an associated 2-Hessian curve that intersects it in itssextactic points. In this paper we fix an error in Cayley's calculations andprovide the correct defining polynomial for the 2-Hessian. In addition, wepresent a formula for the number of sextactic points on cuspidal curves and tiethis formula to the 2-Hessian. Lastly, we consider the special case of rationalcurves, where the sextactic points appear as zeros of the Wronski determinantof the 2nd Veronese embedding of the curve.

The 2-Hessian and sextactic points on plane algebraic curves