Report on the Theory of Numbers.-Part IV. By H. J. STEPHEN SMITH, M.A., F.R.S., Savilian Professor of Geometry in the University of Oxford. 105. General Theorems relating to Composition.-The theory of the composition of quadratic forms occupies an important place in the second part of the 5th section of the Disquisitiones Arithmetica,' and is the foundation of nearly all the investigations which follow it in that section. In accordance with the plan which we have followed in this portion of our Report, we shall now briefly resume the theory as it appears in the Disquisitiones Arithmeticæ,' directing our special attention to the additions which it has received from subsequent mathematicians. We premise a few general remarks on the Problem of composition. If F, (x,, x,,... x) be a form of order m, containing n indeterminates, which, by a bipartite linear transformation of the type x=2αa, B, y Yẞy, a=1, 2, 3,... n, B=1, 2, 3,... n, y=1, 2, 3,... n,. is changed into the product of two forms F(y,, 2, .. Yn) and F,(z1, 229 of the same order, and containing the same number of indeterminates, F, is said to be transformable into the product of F, and F,; and, in particular, if the determinants of the matrix 2 2 3 1 3 which is of the type nxn2, be relatively prime, F, is said to be compounded of F, and F. Adopting this definition, we may enunciate the theorem- —"If F. be transformable into F, XF,, and if F, G, G, be contained in G,, F, F, respectively, G, is transformable into G, XG,; and, in particular, if F, be compounded of F, and F, and the forms F, G, G, be equivalent to the forms G1, F, F, respectively, G, is compounded of G, and G.' 3 2 3 2 2 It is only in certain cases that the multiplication of two forms gives rise to a third form, transformable into their product. Supposing that F, and F, are irreducible forms, i. e. that neither of them is resoluble into rational factors, let I, I, I, be any corresponding invariants of F1, F2, F., and let us represent by B and C the determinants The transformation of F, into F,× F, then gives rise to the relations i denoting the order of the invariants I, I, I,. If one of the two numbers I, and I, be different from zero, we infer that m is a divisor of n. For if "be the fraction reduced to its lowest terms, the equations imply that F, and F, (cleared of the greatest numerical divisors of all their terms) are perfect powers of the order μ; i. e., μ=1, or m divides n, since F and F, are by hypothesis irreducible. We thus obtain the theorem (which however applies only to irreducible forms having at least one invariant different from zero)" No form can be transformed into the product of two forms of the same sort, unless the number of its indeterminates is a multiple of its order." For example, there is no theory of composition for any binary forms, except quadratic forms, nor for any quadratic forms of an uneven number of indeterminates. Again, when m is a divisor of n, let n km, and let b, c, d,, d, represent the greatest numerical divisors of B, C, F, F, respectively; we find The first two of these equations show that the invariants of the three forms F1, F, F, are so related to one another, that we may imagine them to have been all derived by transformation from one and the same form (see Art. 80); the last two (which, it is to be observed, present an ambiguity of sign mi when is even) show that the forms B and F,*, C and F*, are respectively n identical, if we omit a numerical factor. Lastly, let ,,,, be any corresponding covariants of F1, F2, F. The relation of covariance gives rise to the equations where p and q are the orders of the covariants in the coefficients and in the indeterminates respectively. Combining with these equations the values of 2 2 m 3 2 m B and C already given, we see that × F, and × F," are identical, excepting a numerical factor; i. e. that, and, are either identically zero, or else numerical multiples of powers of F, and F. If therefore two forms can be combined by multiplication so as to produce a third form transformable into their product, their covariants are all either identically zero or else are powers of the forms themselves. There is, consequently, no general theory of composition for any forms other than quadratic forms, because all other sorts of forms have covariants which cannot be supposed equal to zero, or to a multiple of a power of the form itself, without particularizing the nature of the form. And even as regards quadratic forms, we may infer that composition is possible only in cases of continually increasing particularity, as the number of indeterminates increases. 106. Composition of Quadratic Forms.—Preliminary Lemmas.-The following lemma is given by Gauss as a preliminary to the theory of the composition of binary quadratic forms (Disq. Arith., art. 234) :— (i.) "If the two matrices in which the sign of equality refers to corresponding determinants in the two matrices; and if the determinants of admit of no common divisor beside a in which the sign of equality refers to corresponding constituents in the two matrices, is always satisfied by a matrix || of the type 2x2, of which the determinant is k, and the constituents integral numbers."* The subsequent analysis of Gauss can be much abbreviated if to this lemma we add three others. * For a generalization of this theorem, see a paper by M. Bazin, in Liouville, vol. xix. p. 209; or Phil. Trans. vol. cli. p. 295. |