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whose coefficients are arbitrary variables, be transformed by the group A of all linear substitutions on xi, x2, whose coefficients are least positive residues modulo p, a prime number, there is brought into existence an infinitude of rational integral functions of ao, *-* *, am, x1, X2, which are invariants under the group. Whether this infinite system possesses the property of finiteness, in general, is an unsolved problem, but in this paper I show that, when the modulus is 2, the system of covariants of a cubief3 is finite and that the fundamental set consists of twenty quantics. This system of covariants, five of which are pure invariants, is derived in explicit form. The methods of generation and proof of the completeness of the fundamental set are developed from the point of view emphasized in a paper on the formal modular invariant theory, by the present writer, in volume 17 of these T r a n s a c t i o n s . t These methods presuppose a knowledge of a fundamental system of formal seminvariants of the given ground form; but this seminvariant system has been given previouslyt for f3 and the modulus 2, by Dickson.

A fundamental system of formal covariants modulo 2 of the binary cubic