The present paper owes its origin to an attempt to solve the problem of Hurwitz in the case of the binary cubic form, viz., to find a set of formal modular invariants of the binary cubic form such that all others could be expressed as polynomials with integral coefficients with these as arguments. In this attempt I have not yet been successful, but methods have been developed which will aid in the solution of the problem and which have resulted in the solution of the similar, though less difficult problem of the determination of a fundamental system of formal modular seminvariants of the binary cubic form, modulis 5 and 7. These methods are so general in their nature that they could easily be used in the determination of a like system with respect to any prime modulus greater than three. The problem of a fundamental system of the seminvariants of the cubic here solved in the case of the moduli 5 and 7 has already been solved by Dickson in the case of the modulus 5 and the methods which he used in that case could no doubt have been used for its solution in other cases. Whatever interest may attach to the present paper does not then lie in the fact that a solution exists nor yet in the fundamental system exhibited, but in whatever power or generality the methods used in the solution may have, and in the fact, hitherto unknown, that in the case of the cubic the number of members of a fundamental system is a function of the modulus instead of being a constant as in the case of the quadratic. The method of annihilators, fundamental in the classical theory of invariantive concomitants, seems at first to be almost useless in the formal modular case. This is due to the fact that not only are these annihilators not linear but they are of an order which depends upon the modulus. Their complexity does indeed make the computation of seminvariants and invariants through their use very laborious in any except the simplest cases, but they furnish simple proofs of general theorems of far reaching significance. * Presented to the Society, October 30, 1920. 56