Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/678

620 NUMBERS 1 x then conversely -^ suppose, y = -(-yx + ay ),=y x where a , /? , y , 8 are integers ; and Ave have, moreover, that is, a 8 - /3 Y = + 1 or 1 according as a8 /3y is = + 1 or - 1. The two forms (a, b, c), (a , b , c) are in this case said to be equivalent, and to be, in regard to the particular transformation, equivalent properly or improperly accord ing as aS - /3y( = a 8 - (3 y) is = + 1 or = - 1. We have, therefore, as a condition for the equivalence of two forms, that their determinants shall be equal ; but this is not a suffi cient condition. It is to be remarked also that two forms of the same determinant may be equivalent properly and also improperly ; there may exist a transformation for which a8-/3y is = +1, and also a transformation for which aS - (3y is = 1. But this is only the case when each of the forms is improperly equivalent to itself ; for instance, a form x 1 Dy 2 , which remains unaltered by the change x, y, into x, y (that is, a, /?, y, 8 = 1, 0, 0, - 1, and therefore aS - /3y = - 1), is a form improperly equiva lent to itself. A form improperly equivalent to itself is said to be an ambiguous form. In what follows, equivalent means always properly equivalent. 27. Forms for a given determinant classes, &c. In the case where D, = 6 2 - ac, is a square the form (a, b, c}(x, y) 2 is a product of two rational factors ; this case may be ex cluded from consideration, and we thus assume that the determinant D is either negative, or, being positive, that it is not a square. The forms (a, b, c) of a given positive or negative determinant are each of them equivalent to some one out of a finite number of non-equivalent forms which may be considered as representing so many distinct classes. For instance, every form of the determinant - 1 is equivalent to (1, 0, 1), that is, given any form (a, b, c) for which b 2 - ac = - 1, it is possible to find integer values a, j3, y, 8, such that a8 - (3y = + 1, and (a, b, c)(ax + (By, yx + S^) 2 = (1,0,1 )(x, y) that is, = x 2 + y*. Or, to take a less simple example, every form of the deter minant - 35 is equivalent to one of the following forms : (1, 0, 35), (5, 0, 7), (3, 1, 12), (4, 1, 18),- (2, 1, 8), (6, 1, 6) ; for the first six forms the numbers a, 26, c have no common factor, and these are said to be properly primitive forms, or to belong to the properly primitive order ; for the last two forms the numbers a, b, c have no common factor, but, a and c being each even, the numbers a, 26, c have a common factor 2, and these are said to be improperly primitive forms, or to belong to the improperly primitive order. The properly primitive forms are thus the six forms (1, 0, 35), (5, 0, 7), (3, 1, 12), (4, 1, 18); or we may say that there are represented hereby six properly primitive classes. Derived forms, or forms which belong to a derived order, present themselves in the case of a determinant D having a square factor or factors, and it is not necessary to consider them here. It is not proposed to give here the rules for the deter mination of the system of non -equivalent forms ; it will be enough to state that this depends on the determination in the first instance of a system of reduced forms, that is, forms for which the coefficients , 6, c, taken positively satisfy certain numerical inequalities admitting only of a finite number of solutions. In the case of a negative de terminant the reduced forms are no two of them equivalent, and we thus have the required system of non-equivalent forms ; in the case of a positive determinant, the reduced forms group themselves together in periods in such wise that the forms belonging to a period are equivalent to each other, and the required system of non-equivalent forms is obtained by selecting one form out of each such period. The principal difference in the theory of the two cases of a positive and a negative determinant consists in these periods ; the system of non-equivalent forms once arrived at, the two theories are nearly identical. 28. Characters of a form or class division into genera. Attending only to the properly primitive forms for in stance, those mentioned above for the determinant - 35 the form (1, 0, 35) represents only numbers / which are residues of 5, and also residues of 7 ; we have, in fact, f=x 2 + 35y 2 , =x^ (mod. 5), and also = .r 2 (mod. 7). Using the Legendrian symbols (^j and (^j, we say that the form (1, 0, 35) has the characters ( - ) , K ) = + + . o/ i / Each of the other forms has in like manner a determinate character + or - in regard to ( - ) and also in regard to A = ] : and it is found that for each of them the characters are + + or else - - (that is, they are never H or H ). We, in fact, have (iXf) (1, 0, 35) + + (4. 1, 9) (5, 0, 7) - - (3, 1, 12) and we thus arrange the six forms into genera, viz., we have three forms belonging to the genus (V )> r* ) = + + > /* J - = - - , these char and three to the genus (-= 5 acters + 4- and - - of genera being one-half of all the combinations ++, , + , +. The like theory applies to any other negative or positive determinant ; the several characters have reference in some cases not only to the odd prime factors of D but also to the numbers 4 and 8, that is, there is occasion to consider also the Legendrian symbols [], = (-!) , and /2 J(/ 2 -i) ^ * . - ( 1) , and there are various cases to be con- sidered according to the form of D in regard to its simple and squared factors respectively ; but in every case there are certain combinations of characters (in number one-half of all the combinations) which correspond to genera, and the properly primitive forms belong to different genera accordingly, the number of forms being the same in each genus. The form (1, 0, - D} has the characters all +, and this is said to be the principal form, and the genus containing it the principal genus. For a given determinant, the characters of two genera may be compounded together according to the ordinary rule of signs, giving the char acters of a new genus ; in particular, if the characters of a genus are compounded with themselves, then we have the characters of the principal genus. 29. Composition of quadratic forms. Considering X, Y, as given lineo-linear functions of (x, y), (x, y ), defined by the equations X=p ff xx +p l xy +p 2 ijx +p$y , Y= q<?cx + qjxy + q 2 yx + qgjy , the coefficients p , p v p 2 , p 9 ? , q v q 2 , q y may be so^ con nected with the coefficients (A, , C), (a, b, c), (a, 6 , c ), of three quadratic forms as to give rise to the identity (A, , C)(X, Y}"-=(a, b, e)(x, 7/) 2 . ( , b , c )(x , y f; and, this being so, the form (A, B, (7) is said to be com pounded of the two forms (a, 6, c) and (a , b , c ), the order of composition being indifferent.