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It is to be expected that the systematic introduction of irrational covariants will enrich geometry with curves and surfaces, not previously observed or discussed, allied to any given fundamental system by projective relations. Well known irrationalities also must be expected often to reappear. In 1888 Dr. HILBERT t remarked the existence of two nets of conics covariantly related to a general curve of third order in a plane. Two additional systems, nets of curves of the second class, can easily be defined by anl equation closely analogous to Hilbert's. These systems of conics and the four cubics whose polars they are prove to be not entirely unknowvn hitherto, for their dually equivalent loci and envelopes form the basis of the one-to-one correspondences upon the point-cubic and the line-cutbic respectively. That only two of every three such correspondences are found here is due to the domain of rationality that is assumed. By employing the irrationality that occurs in HESSE'S canonical fornm of the cubic I ani able to identify Hilbert's two systems of irrational covariant conics and to exhibit their relation to the other two systems just mentioned. As a consequence it is found possible to give explicitly covariant equations of definition for the two cubics which have the same Hessian and for those which have the same Cayleyan as a given fundamental cubic. These results are here derived by the aid of a canonical form of the cubic containing Hesse's irrationality. The desirable invariantive proofs of these results are given by Professor GORDAN in a paper presented by him to the Americani Mathematical Society for publication in its Transactions.4

Conics and cubics connected with a plane cubic by certain covariant relations