greatest multiple of [A] not surpassing s/~D. Similarly (2B, (2&+1) B, C) is equivalent to (2B, B, C), and is reduced if (2&+1) B be positive, and be the greatest uneven multiple of [B] not surpassing >J~D. There are, therefore, as many reduced ambiguous forms as there are forms in CI; and there are no more, because it is readily seen that every reduced ambiguous form is included in ono or other of the two series of forms (A, IcJL, C) and (2B, (2fc+l)B, C) which we have obtained. But every ambiguous class contains two reduced ambiguous forms (Art. 94); we infer, therefore, that for positive as well as for negative determinants the number of ambiguous classes i3 one-half the number of the forms CI, i. e. one-half of the number of assignable generic characters. Combining this result with the theorem at the commencement of this article, we obtain a proof of the impossibility of at least one-half of the assignable generic characters. As this proof is independent of the law of quadratic reciprocity, we may employ the result to demonstrate that law. [Gauss's second demonstration, Disq. Arith., art. 262.] Let p and q be two primes, and first let one of them, as p, be of the form 4n+l. If s=—1, we infer that ^p =—1; for if (^ = +^> we should have w'sp, mod. q, and consequently there would exist a form ^q, u, -—of det. p, of which the character would be ^0=—1, i. e. there would be 2 genera of forms of determinant p. Similarly, if (~~)= we have o»'= +q, moi.p; and (p, u, - is a form of det. ±q. If ±q be of the form 4»+l, P / there will be but one genus of forms, i.e. the principal genus; whenco ^-^= These two conclusions are sufficient to establish the theorem of reciprocity when one of the two primes is of the form 4n-f-l. If both p and q be of the form 4n+3, there are four assignable characters for the determinantpq. Of these ^-^=1, ^/^=1; —1> y — ^» are P03~ sible, as is shown by the existence of the forms (1, 0, —pq), (—1, 0,pq); the other two are therefore impossible. Hence in the form (p, 0, — q) we must have either (j-^ = ^= (~^)' 01 (^) = _*=0l^0' eX~ presses the theorem of reciprocity for this case. The supplementary theorems relating to 2 and — 1 can be similarly proved. 116. Equality of the Number of Genera and of Ambiguous Classes.— In the preceding article it has only been shown that h cannot exceed h. But, as we have already seen (Art. 102) that the number of actually existing genera is one-half the whole number of assignable generic characters, we know that Jc=h. To prove this, by the principles of the composition of forms, it is sufficient to show that n=n', i. e. that the problem "to find a class which by its duplication shall produce a given class of the principal genus " is always resoluble. This problem Gauss actually solves (Disq. Arith., art. 286, 287); he shows, first, that any proposed binary form, belonging to the principal genus of its own determinant, can be represented by the ternary quadratic form X1—2TZ; and, secondly, that from this representation we can always deduce a binary form, which shall produce by its duplication the proposed form. This solution implies a previous investigation of the theory of ternary quadratic forms, and cannot be properly introduced here. A more elementary method, however, has been given by M. Arndt (Crelle, lvi. p. 72). Let D=AS", 8J representing any square dividing D; M. Arndt observes that the ratio of the number of actually existing genera to the whole number of assignable generic characters is the same for each of the two determinants D and A. To prove this we make use of the following subsidiary proposition:— "If f=(a, b, o) be a properly primitivo form of any det. D, and if 8M and 6 be two numbers relatively prime, the necessary and sufficient condition for the resolubility of the congruence ax,+2bxy+cy'~e, mod 831 (A) is that the supplementary characters of / (if any), and the particular characters of f (if any) which relate to uneven primes dividing both M and D, should coincide with the corresponding characters of 9." We may add (though this is not necessary for our present purpose), that if 9, and 92 be two values of 9 for each of which the congruence (A) is resoluble, it- is resoluble for each an equal number of times. On reference to the Table in Art. 98, it will be seen that the particular characters proper to the determinant A are included among the particular characters proper to D. Let then (r) and (r, V) represent any two complete generic characters for the determinants A and D, the particular characters common to the two complete characters having the same values attributed to them in each. It may then be shown that the genus (r, V) is or is not an existent genus, according as (r) is or is not existent. For (1) if (r, r') be actually existent, let 9 be a number prime to 2D and capable of primitive representation by some class of that genus; the congruence u'gD, mod. 9 is therefore resoluble; i.e. the congruence <u2=A, mod. 9, is resoluble, so that 9 can be represented by a class of properly primitive forms of det. A, or the genus (D is actually existent. And (2) if (r) be an existing genua, let f be a form included in (r), and 9 a number prime to 2D and satisfying the generic character (r, I"). It appears from the subsidiary proposition that some number 0 of the linear form 8mD + 9 is capable of representation 'by /; if & be the greatest common divisor of the indeterminates in the representation of 0 by /, the congruence us=A, and consequently the congruence <i>'=D, is resoluble for the modulus T; i.e. —, the character of which coincides ... • o o with the character of 9, and therefore with that of the genus (r, r')> Is capable of representation by a form of det. D, or (r, r') is an actually existing genus. If, then, K be the number of particular characters contained in (r, r') »J>4 not in (r), the numbers of actually existing genora and assignable generie characters for the det. D are each 2» times the corresponding numbers for the det. A. It appears from this result that it will be sufficient for our present purpose to consider determinants not divisible by any square. If (a, 6, e) be a form of the principal genus of such a determinant (we suppose that a is prime te D), the equation <l»5^-2&M/-f.<sy,■=w, is resoluble with values of u prime to D; for if a^a'b*, 3* representing the greatest square divisor of a, the equation is certainly resoluble in relatively prime integers, by virtue of a celebrated theorem of Legendre#; and the values of f which satisfy it are prime to D; whence, if x=p - ■, y=fin, <■»=/* -g, p denoting a multiplier, which renders the values of x, y, and u integral and relatively prime, the equation ax' + 2bxy+cy2=ii>' will be satisfied, and the values of w will be prime to D. The form (a, b, e) is therefore equivalent to a form of the type (w2, X, v); and this form is produced by the duplication of (u>, X, vw) if w be uneven, and of (2w, X-f oi, v') if ci> bo even. 117. Arrangement of the Classes of the principal Genus.—If C be a class of the principal genus, the classes C, Ca, C*,. . . will all belong to that genus. And it will be found, by reasoning similar to that employed in Baler's second proof of Format's theorem (see Art. 10 of this Report), (1) that the classes 1, C, (?,,.. are all different until we arrive at a class (>, equivalent to the principal class; (2) that fi is either equal to, or a divisor of, the number n of classes in the principal genus; (3) that if 0=1, r is a multiple of p. The fi classes C, Ca, C3, . . . O-1, 1, are called the period t of the class C; C is said to appertain to the exponent ft; and the determinant is regular or irregular according as classes do or do not exist which appertain to the exponent n. With the former case we may compare the theory of the residues of powers for a prime modulus; with the latter the same theory for a modulus composed of different primes (see Art. 77). (i.) When the determinant is regular, we may take any class appertaining to the exponent n as a basis, and may represent all the classes of the principal genus (to which we at present confine ourselves) as its powers. It will then appear (1) that if d be a divisor of n, the number of classes appertaining to the exponent d is \L (d); so that, for example, the number of classes that may be taken for a base is yp (n): (2) that if ef=n, the equation X"=l will be satisfied by e classes of the principal genus; and if these classes be represented by A„ A,, . . . A,, each of the equations X/=A will be satisfied by f different classes of the same genus: (3) that the only classes of the principal genus which satisfy the equation X*=l are those which satisfy the equation X'e=l, where d is the greatest oommon divisor of k and n. It will be seen in particular mat the equation X*=l admits of only one, or only two solutions, according as n is uneven or even; t.«. the principal genus of a regular determinant cannot contain more than two ambiguous classes. To obtain a class appertaining to the exponent n, Gauss employs the same method which serves to find a primitive root of a prime number (Art. 13; Disq. Arith., art. 73, 74), and which reposes on the observation, that if A and B be two classes appertaining to the exponents a and /3, neither of which divides the other, and if M, the least common multiple of a and /?, be resolved into two factors p and q, relatively prime and such that p divides a and q divides /3, the class A? x B« will appertain to the exponent M. (ii.) When the determinant is irregular, the classes of the principal genus * Theorie de* Nombres, ed. 3, vol. i. p. 41; piisq. Arith., art. 294. t These periods of non-equivalent classes are not to be confounded with the periods of equivalent reduced forms of Art. 98. cannot be represented by the simple formula C*. and we must employ an expression of the form C,'1 x C^1 x C4, .... To obtain an expression thus representing all the classes of the principal genus, we take for C, a class appertaining to the greatest exponent 6\ to which any class can appertain; and in general for CM we take a class appertaining to the greatest exponent 0,, to which any class can appertain when its period contains no class, except the principal class, capable of representation by the formula C, x C, x .. / n C^-i The number K=92 X 0, X .. . is called by Gauss the exponent of H n irregularity; and similarly we might term r—5-, „ „ „, &c, the second, third, &c, exponents of irregularity. From the mode in which the formula C,'1 x CJ*'J x . . is obtained, it can be inferred that 0, is divisible by 0,, 0a by 0,, and so on; whence it appears that a determinant cannot be irregular unless Ji be a divisible by a square; nor can it have r indices of irregularity unless n be divisible by a power of order r+1. Moreover, whenever the principal genus contains but one ambiguous class, the determinant is either regular or has an uneven exponent of irregularity; if, on the contrary, the principal genus contain more than two ambiguous classes, the determinant is certainly irregular, and the index of irregularity even; if it contain 2r ambiguous classes, the irregularity is at least of order K, and the K exponents of irregularity are all even. A few further observations are added by Gauss. Irregularity is of much less frequent occurrence for positive than for negative determinants; nor had Gauss found any instance of a positive determinant having an uneven index of irregularity (though it can hardly be doubted that such determinants exist). The negative determinants included in tho formulae, —D=216fc+27, = 1000&4-75, =1000&+675, except —27 and —75, are irregular, and have an index of irregularity divisible by 3. In the first thousand there are five negative determinants (576, 580, 820, 884, 900) which have 2 for their exponent of irregularity, and eight (243, 307, 339,459, 675, 755, 891, 974) which have 3 for that exponent; the numbers of determinants having these exponents of irregularity arc 13 and 15 for the second thousand, 31 and 32 for the tenth. Up to 10,000 there are, possibly, no determinants having any other exponents of irregularity; but it would seem that beyond that limit the exponent of irregularity may have any value. 118. Arrangement of the other Genera.—In the preceding article we have attended to the classes of the principal genus only; to obtain a natural arrangement of all the properly primitive classes, we observe that, if the number of genera be 2^, the terms of the product (1 + r,) (1 + r,) (1+rs)... (1 + rM), in which r< represents any genus not already included in the product of the 1— 1 factors preceding 1 will represent all the genera. If, then, A„ A,,... represent any classes of the genera r„ ra,. . rM respectively, and |C| be the formula representing all the classes of the principal genus, the expression |K|=|C I x(l+A,) (1+A2) . . . (1+AM) supplies a type for a simple arrangement of all the classes of the given determinant. When every genus contains an ambiguous class, it is natural to take for A,, Aa,. . AM, the ambiguous classes contained in the genera r„ ra, . . 1^ respectively. When the principal genus contains two ambiguous classes (and when, consequently, one-half of the genera contain no such classes), let C, be the class taken as base (or, if tho determinant be irregular, as first of the bases) in the arrangement of the classes of the principal genus, and let Cl'=Cl; it may then be shown that Q, will belong to a genus containing no ambiguous class, and that the formula |K|=|C| X (1+Q,) (1+A2) . . . (1+A^), in which A,, . . A^, are ambiguous classes, represents all the classes *. In general, if the principal genus contain 2* ambiguous classes (a supposition which implies that the determinant is irregular, having «c even exponents of irregularity, and that there are only 2*_" -genera containing ambiguous classes)—let Cl'=Ct; 02,=CJ; . . .Cl/=Ct—it will be found that all the classes are represented by the formula |K|=|C| x(l + 0,) (l+Oa). . (1+Oc) (1 + A.c+i) • • (l+AM),in which Ae+i,. . . AM are ambiguous classes, and fl,, Cil. . . £1K classes belonging to genera containing no ambiguous class f. A similar arrangement of the improperly primitive classes (when such classes exist) is easily obtained. Let 2 denote the principal class of improperly primitive forms, i. e. the class containing the form ^2,1, — we have seen (Art. 113) that the number of properly primitive classes which, compounded with 2, produce 2, is either one or three. When there is only one such class, the number of improperly primitive classes is equal to that of properly primitive classes; and if |K| be the general formula representing the properly primitive classes, the improperly primitive classes will be represented by 2 X |K|. When there are three properly primitive classes, which, compounded with 2, produce 2, the principal class will be one of them, and if <p be another of them, <p? will be the third; also & and will belong to the principal genus, and will appertain to the exponent 3. When the determinant is regular, instead of the complete period of classes of the principal genus, 1, C, C, . . C"-1, wo take the same series as far as the class C** exclusively; when the determinant is irregular, we can always choose the bases C„ Ca,. . in such a manner that the period of one of them shall contain A and <f, and this period we similarly reduce to its third part by stopping just before we come to <p or <j>2. Employing these truncated periods, instead of the complete ones, in the general expression for the properly primitive classes, we obtain an expression, which we shall call |K'|, representing a third part of the properly primitive classes, and such that 2 X |K'| represents all the improperly primitive classes. 119. Tabulation of Quadratic Forms.—In Crelle's Journal, vol. lx. p. 357, Mr. Cayley has tabulated the classes of properly and improperly primitive forms for every positive and negative determinant (except positive squares) up to 100. The classes are represented by the simplest forms contained in them J; the generic character of each class, and, for positive determinants, the period of reduced forms (Art. 93) contained in it, are also given. The "Gauss employs a class Q, producing C, by its duplication, both when one and when two ambiguous classes are contained in the principal genus. The number of classes requisite for the construction of the complete system of classes is therefore p in either case, since C, may be replaced by Q3,. t The principles employed by Gauss for the arrangement of the classes of a regular determinant are extended in the text to irregular determinants. If the determinant hare c' uneven exponents of irregularity, the number of classes requisite for the construction of the complete system of classes is /t+ic'. X The simplest form contained in a class is that form which has the least first coefficient of all forms contained in the class, and the least second coefficient of all forms contained in the class and having the least first coefficient. If a choice presents itself between two numbers differing only in sign, the positive number is preferred. In the case of an ambiguous class of a positive determinant, the simplest ambiguous form contained in the class is taken as its representative. |