representation of a binary form ƒ of determinant D by a ternary form of determinant 1 to be, that ƒ should be a form of the principal genus; or, if D=1, mod 4, that ƒ should be a form either of the principal genus, or else of that genus which differs from the principal genus only in having the character y=—1, instead of y=+1. Again, because the reduction of Lagrange is applicable to complex binary forms, the reduction of Gauss is applicable to complex ternary forms. It is thus found that the number of classes of such forms of a given determinant is finite; and in particular that every form of determinant 1 is equivalent to one or other of the forms —x2 —y2 —z2 and x2+iy2+iz2, of which the former cannot represent numbers=i, or=1+i, mod 2; and the latter cannot primitively represent numbers=2, or=2(1+i), mod 4. The method of reduction itself sup * If F=ax2+a'y2+a′′z2+2byz+2b'xz+26′′xy is a ternary form of determinant ▲, and Ax2+A'y2+A′′z2+2Byz+2B'xz+2B"xy its contravariant, by applying the reduction of Lagrange to the form ax2+2b′′xy+a'y2, we can render N.a≤2√N. A′′ (Dirichlet in Crelle's Journal, vol. xxiv. p. 348); and by applying the same reduction to the form A'y2+2Byz+A′′z2, we can render N. A"≤2N. aA. The reduction of Gauss consists in the alternate application of these two reductions until we arrive at a form in which we have simultaneously N. a≤2/N. A′′, N. A′′≤2√√Ñ.as; and consequently N. a4 N.A, N.A"≤43/N. 4. If A=1, we have N.a≤4, N.A"≤4; whence a and A′′ can only have the values 0, ±1, ±i, ±(1+i), ±(1−i), ±2, +2i; and it will be found, on an examination of the different cases that can arise, that the reduction can always be continued until a and A" are either both units, or both In the former case, by applying a further transformation of the type zero. 1, μ”, μ' 0, 1, μ the coefficients b, b', b' may be made to disappear; and we obtain a form equivalent to F, and of the type ex2+e'y2+e"z2, e, e', e" representing units of which the product is -1. In the latter case the form obtained by applying the reduction of Gauss is of the type a'y2+a"z2+2byz+2b'xz; whence a'b'=1, so that b' is a unit which we shall call e; and the form e2y? +a'z2+2byz+2exy, by a transformation of the type 1, 0, μ' 0, 1, μ 0, 0, 1 is changed into one of the four forms 2y2+2exz, e2y2+z2+2exz, e2y2+iz2+2exz, e3y2+(1+i)z2+2arz; of which the first two by the transformations are changed into x2+iy2+iz2. (See Disq. Arith. art. 272–274.) lies a transformation of any given form of determinant 1 into one or other of those two forms. If D=i, or 1+i, mod 2, no binary form of determinant D can be reresented by -2-y2-22, because D cannot be represented by the conravariant of that form, i. e. by the form --y- itself. Consequently, if D=i, or 1+i, mod 2, the binary forms of its principal genus re certainly capable of primitive representation by x2+iy2+iz2. If D=1, mod 2, no form of the principal genus can be primitively represented by x2+iy2+iz. Let f (a, b, c) be such a form, and let us suppose, as we may do, that b is even, so that ac=1, mod 2, and c=1, mod 2 (the supposition a=ci is admissible, because ƒ is of the principal genus); if possible, let the prime matrix α,β (of which A, B, C are the determinants) transform x2+iy2+iz2 into ƒ; we have the equations a=a2+ia12+ia"2, c=ß2+iß12+iß", D=A2—¿B2—iC3, from which, and from the congruences D=a=c=1, mod 2, we infer the incompatible conditions a'+ia"=B'+iẞ"=0, mod 1+i, A=1, mod1+i; i.e. fis incapable of primitive representation by 2+iy2+iz2. If, therefore, D=1, mod 2, the forms of its principal genus are capable of primitive representation by -x2-y2-22. We may add that when D=±1, mod 4, the forms of that genus which differs from the principal genus only in having the character y=-1, instead of y=+1, are capable of primitive representation by x2+iy2+iz, but not by x2-y2-23. Lastly, let D=0, mod 2. If D=2, or=2(1+i), mod 4, D cannot be primitively represented by 2-iy-iz2, the contravariant of x2+iy2+iz2; i.e. no form of determinant D can be primitively represented by x2+iy+iz2; >so that forms of the principal genus are certainly capable of primitive representation by --y-z2. But if D=2i, or= 0, mod 4, the forms of the principal genus are capable of primitive representation by both the ternary forms-y-22 and x2+iy2+iz. For if f=(a, b, c) be a form of the principal genus of any even determinant, ƒ can only represent numbers=0, or=1, mod 2; so that a ternary form of determinant 1 and of the type f+p"z2+2qyz+2q'xz will be equivalent to —x2-y2-z2, or to x2+iy2+iz2, according as p"=0, or=1, mod 2, on the one hand, or p"=i, or=1+i, on the other hand. Again, if (k, k') is a value of the expression (a, b, c), mod D, (in D which we now suppose a uneven and 6 semieven or even), (+ 11++, *') is another value of the same expression; and it can be shown* that when * - If f+p"z2+2qy2+2qxz is a ternary form of det. 1, derived from the value (k, k') of the expression (a, b, c), mod D, k is the coefficient of yz in the contravariant form. Hence a=k2—D(q′′2—ap′), or ap′′=q2+ a D Observing that a=1, mod 2 D=2i, or 0, mod 4, one of the two forms of determinant 1, and of the type f+p"2+2qye+2q'xz, which are deducible by the method of Gauss from those two values, satisfies the condition p"=0, or=1, mod 2, while the other satisfies the condition p"-i, or 1+i, mod 2; that is, f is capable of primitive representation by both the forms -x2-y2-z2 and x2+iy2+iz3. The preceding theory supplies a solution of the problem, "Given a form of the principal genus of forms of determinant D, to investigate a form from the duplication of which it arises." Let f=(a, b, c) be the given form, and let us suppose (as we may do) that a and c are uneven. When D=i, or 1+i, mod 2, let be a prime matrix (of which the determinants are A, B, C) transforming x2+iy2+iz2 into (a, —b, c); and let represent the binary form (C-iB, A, iC-B); then the matrix - 'ß'+iß", ß, ß, —¿(ß'—¿ß'')' (Z) transforms ƒ into ×*; and is a prime matrix, for its determinants C-iB, 2A, and iC-B are not simultaneously divisible by any uneven prime (because A, B, and C are relatively prime), and are not simul g=0, or 1, mod 2, we see that p′′=0,1, or=i, 1+i, mod 2, according as gruous to 1+i, mod 2, if D=0, mod. 4, and to i, mod 2, if D=2i, mod 4, since k is evidently uneven in either case. then a D 2 =i, 1+i, mod 2; that is, in one of the two forms f+p"z2+2qyz+ 2q'xz, p=0, or 1, mod 2, and in the other p", or 1+i, mod 2. (I12-Lolз) (Poxx' +P2xy′+P2x'y+Pzx'y')2 +(P1 P2 PoP3) (q。xx′+q1xy′+Q¿x'Y+qzx'y')3 = [(Pol2-P2lo)2+(Pol3-Palo+P12-P21) x'y' +(P193-P381)y] [(Poli-P1o)2+(Pol3-P340+P2q1-P12)x+(P2s-Pala)y2]; in which we have to replace the quantities Po P1 P2 P3 20 91 92 la by the elements of the matrix (Z). taneously divisible by 1+i, because (Z) is congruous, for the modulus 1+i, to the first or second of the matrices 0, (0, 1, 1, 9) and (1, 0, 1, 0) 1, 0, 0, according as a=i, c=1, or a=1, c=i, mod 2. Consequently (Z') is a form the duplication of which produces f. When D=1, or=0, mod 2, let the prime matrix transform -- y2 —z2 into (a, b, c). As we cannot have simultaneously a=ß, a'=ß', a"=ẞ", mod (1+i), we may suppose that a and ẞ are incongruous, mod (1+i). If p=(B+iC, iA, B—¿C), the matrix transforms into xp, and is a prime matrix, being congruous to one or other of the matrices (Z') for the modulus 1+i, in consequence of the two suppositions that a and c are uneven, and that a and ẞ are incongruous, mod (1+i): so that ƒ arises from the duplication of p. From the resolubility of this problem we can infer (precisely as Gauss has done in the real theory) that that half of the assignable generic characters which is not impossible corresponds to actually existing genera. We can also deduce a demonstration of the theorem that any form of determinant D can be transformed into any other form of the same genus, by a transformation of which the coefficients are rational fractions having denominators prime to 2D. For every form which arises from the duplication of an uneven primitive form—that is, every form of the principal genus-represents square numbers prime to 2D, and is therefore equivalent to a form of the type (x2, But (1, 0, -D) is transformed ; i. e. any two forms of the principal genus can be transformed into one another by transformations of the kind indicated. Again, if f,, f, be two forms of any other genus, a form of the principal genus exists satisfying the equation f=4xf. But since can be transformed into the principal form, we can assign to the indeterminates of rational values, having denominators prime to 2D, which shall cause φ to acquire the value + 1; and thus, from the transformation of f, into f, × ¢, we deduce a rational transformation of f, into f1, the coefficients of which have denominators prime to 2D. The truth of the converse proposition, "Two forms which are transformable into one another by rational transformations having denominators prime to 2D belong to the same genus," is evident from the definition of the generic characters themselves. The proposition itself is of some importance, as it furnishes a verification of the completeness of the enumeration of generic characters contained in Table III. II. “Inquiries into the National Dietary." By Dr. E. SMITH, F.R.S. Received April 28, 1864. (Abstract.) The paper contains an abstract of the scientific results of an inquiry which the author had undertaken for the Government into the exact dietary of large classes of the community, viz. agricultural labourers, cotton operatives, silk-weavers, needlewomen, shoemakers, stocking-weavers, and kidglovers. The inquiry in reference to the first class was extended to every county in England, to North and South Wales and Anglesea, to the West and North of Ireland, and to the West, North, and part of the South of Scotland, whilst in reference to the other classes it was prosecuted in the towns where they were congregated. The object of the investigation was to ascertain in the most careful manner the kind and quantity of food which constitutes the ordinary dietary of those populations; and the inquiry was in all cases made at the homes of the operatives. The number of families included in the inquiry was 691, containing 3016 persons then living and taking food at home. The calculations of the nutritive elements are made upon the basis of an adult, two persons under the age of 10 and one over that age being regarded as an adult, and of the elements, the carbon and nitrogen are calculated in each article of food, whilst the free hydrogen is separately estimated as carbon upon the total quantities. The author then cites the estimations which in his papers in the Philosophical Transactions for 1859 and 1861 he had made of the quantity of carbon and nitrogen emitted by the body under various conditions, and computes on those bases the amounts of those substances which are required as food by various classes of the community. He then proceeds to state the quantities which have been actually found in the dietaries of the persons included in this investigation, and the great variations which the inquiry had brought to light. He also compares the nutriment with the cost of it in the food, and states the proportion which the nitrogen bears to the carbon in each of the classes and in the different localities. Each article of food is then considered separately, and the frequency with which, as well as the average quantity in which, it was obtained by these populations is stated. |