### Extension of two Theorems on Covariants

J. H. Grace
1904 Proceedings of the London Mathematical Society
1. Two of the best known elementary results in invariant algebra are the following:-(i.) The Jacobian of a Jacobian of two binary forms with a third form is reducible. (ii.) The product of two Jacobians can be expressed as an aggregate of products each containing three factors. Involved in each statement is the condition that all the forms considered are of order 2 at least; accordingly, in extending the theorems I shall deal with perpetuants, and here content myself with the remark that the
more » ... remark that the corresponding limitation for the extended theorems is easily discovered. The first theorem has been extended by Jordan,* but, perpetuants not being considered by him, the extended theorem is not in its simplest form. 2. Consider now the first of the two results. Suppressing factors of the type a x , it is equivalent to saying that a symbolical product of the form (ab)(ac) is reducible, or, in other words, that, if an irreducible covariant contain three symbols, it must have at least three factors of the type (ab). The extension is now clear, for it is known that a product containing i symbols is reducible unless it contains at least 2 1 " 1 -1 factors of the determinantal type. But the Jacobian theorem may be stated in a more general form, viz., if a product contain any number of symbols divided into three sets, and the sum of the exponents of factors containing two symbols belonging to different sets be less than three, then the form is reducible. * See Liouville, 1879. Jordan's results in this connection are correct for forms of order not greater than 12, but after this point his upper limit to the order of an irreducible form is too large. I do not know of a case in which the highest order of an irreducible covariant of a system of binary forms is not given by choosing the greatest of the integers n, 2M -2, 3M -6, 4M-14, 5»-30, 6M-62, ... where n is the greatest order of forms in the system. I have examined the cases where n "jp 30, and Mr. A. 1'. Thompson has, I believe, gone considerably further.