On Lists of Covariants

I USE the term covariants to include invariants, and I write particularly concerning lists of covariants (groundforms) of the binary quintic and sextic, those of quantics of lower degree being few and well known. When the weight of a covariant is spoken of in this article, it must be understood to mean the weight of its first term or "source." The symbol 5ft will denote that covariant of the quintic whose weight is n, and 6m that covariant of the sextic whose weight is m. Thus, for example, 52 and 62 represent the hessians (weight 2) of the quintic and sextic respectively. The only case of ambiguity is 615, for which weight there are two covariants : one of these may be denoted by 616a, the other by 6 m . The table printed on the next page exhibits the terminology of different writers. Professor Cay ley's* superb collection of the covariants of the quintic, in wnich each is designated by a letter of the alphabet, is arranged, as will be observed, first according to the degree in the coefficients, and secondly according to the order in the variables. Thus 50, of the first degree, is called A, and 52 and 54, of the second degree, come next ; but 54 being of order 2 while 52 is of order G, the letter B is assigned to 54, and so on. The small italics contained in the column headed by the name of Dr. Salmon f are the symbols used in a table at the end of his work, illustrative of transvection, and denote the seminvariants which form the sources of covariants of the quintic and of higher quantics as well. The letters a, g, h, i, ƒ, &, are therefore used by him also for the sextic, together with I, m, n, q, representing respectively 68, 614, 620, 610. Clebsch J and GroMan § differ but slightly in their nomenclature. Faa de Bruno || designates invariants by the letter 2, with subscripts indicating degree, and other covariants by the letter C\ with subscripts indicating order and degree. The column headed by the name of Professor Sylvester contains his table!" of germs for the quintic, each source having its distinguishing germ, i. e., the coefficient in it of the highest power of the final coefficient of the quintic. Thus, the quintic being