Meteorological Observations made at different Stations in connexion with September 5. EOTAL OBSERVATORY, GREENWICH. Meteorological Observations made at different Stations in connexion with the Balloon Ascent on September 5 {continued). Meteorological Observations made at different Stations in connexion with September 8 (continued). Ko^iL Observatory, Greenwich. 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 Arithmetieee,' 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 Eeport, we shall now briefly resume the theory as it appears in the ' Disquisitiones Arithmetical,' 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, ... #„) be a form of order m, containing n indeterminates, which, by a bipartite linear transformation of the type a = l, 2, 3, ... n, I is changed into the product of two forms Fa(y,, yv .. gn) and F3(21, z2,,.. *B) of the same order, and containing the same number of indeterminates, F, is said to be transformable into the product of F2 and F3; and, in particular, if the determinants of the matrix | a«,P,v |' which is of the type n x n1, be relatively prime, F, is said to be compounded of F2 and F3. Adopting this definition, we may enunciate the theorem—" If F, be transformable into Fa x Fa, and if F,, G2, G3 be contained in G,, F2, F, respectively, G, is transformable into G2 x G3; and, in particular, if F, be compounded of F2 and F3, and the forms F,, Ga, G3 be equivalent to the forms Gj, Fa, F3 respectively, G, is compounded of G2 and G3." It is only in certain cases that the multiplication of two forms gives rise to a third form, transformable into their product. Supposing that Fa and F3 are irreducible forms, t. e. that neither of them is resoluble into rational factors, let I„ Ia, I3 be any corresponding invariants of F,, F3, F3, and let us represent by B and C the determinants dxa and dzy V The transformation of F, into F3 x F3 then gives rise to the relations If!!' I1xB"=IJxF,< mi IlxC"«I,xF,', i denoting the order of the invariants I„ Ia, I3. If one of the two numbers I2 and I3 be diflferent from zero, we infer that m is a divisor of n. For if l- bo the fraction — reduced to its lowest terms, the equations I1"xB",=Ia-xF3'< imply that Fa and F3 (cleared of the greatest numerical divisors of all their terms) are perfect powers of the order p; i.e.,(i=l, or m divides «, since Fa and F3 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 », let n=km, and let 6, e, d3, d3 represent the greatest numerical divisors of B, C, F3, F3 respectively; we find The first two of these equations show that the invariants of the three forms F., Fa, F3 are so related to one another, that we may imagino 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 ■when ^ is even) show that the forms B and F,*, C and Fs*, are respectively identical, if wo omit a numerical factor. Lastly, let *a, #, be any corresponding covariants of F,, F2, Fs. Tho relation of covariancc gives rise to the equations mp-q •,(*„ ^,...^)xB " =*2(y„ y2, .. yj x F/(*„ ... zn) *,(*,. • • *.) x C " -•,(*,. zt, .. O x F*(i/„ ...), where p and q are the orders of the covariants in tho coefficients and in the indeterminates respectively. Combining with these equations the values of 1 «. B and C already given, we see that *2 x F," and *3 x Fam are identical, excepting a numerical factor; t. e. that <l>2 and *3 arc either identically zero, or else numerical multiples of powers of F2 and F3. 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 bo supposed equal to zero, or to a multiple of a power of the form itself, without particularizing the nature of the form. Arid even as regards quadratic forms, we may infer that composition is possiblo 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 I A _| A, A2... A„ in which the sign of equality refers to corresponding determinants in the two admit of no common divisor beside matrices; and if the determinants of A in which the sign of equality refers to corresponding constituents in the two matrices, is always satisfied by a matrix \k\ of the type 2x2, of which the determinant is k, and the constituents integral numbers."* The subsequent analysis of Gauss can bo much abbreviated if to this lemma we add three others. • For a generalization of this theorem, see a paper by M. Bazin, in liourille, Toi. xii. p. 209; or Phil. Trans, vol. cli. p. 295. |