306 REPORT — 1861. 



similar distinction subsists between proper and improper equivalence ; the 

 radical sfD entering with the same sign, or with opposite signs, into the factors 

 •which are transformed into one another, according as the transformation is 

 proper or improper. 



92. Problem of Equivalence — Forms of a Negative Determinant. — To 

 complete the solution of the problem of equivalence, we consider, first, forms 

 of a negative, and then those of a positive and not square determinant. 



A form (a, b, c) of a negative determinant D = — A, which satisfies the 

 conditions enunciated in the following Table, is called a reduced form. The 

 symbols [26] etc. are used to denote the absolute values of the quantities 

 enclosed within the brackets. 



The essential character of a reduced form is sufficiently expressed by the 

 two symmetrical conditions [26] < [a], and [26]<[c]. The third general 

 condition (which combined with the first implies the second), and the special 

 conditions, are, it may be said, artificial restrictions, intended to enable us to 

 enunciate with precision the theorem that "every class contains one, and 

 only one, reduced form." 



To show that one reduced form always exists in any given class, we select 

 from the given class all those forms in which the coefficient of x^ is the least ; 

 and again, from those forms we select that one form, (a, b, c), or those two 

 forms, (a, b, c) and (a, — b, c), in which the coefficient of y^ is the least. The 

 single form (a, b, c), or the two forms (a, b, c), (a, —b, c), thus obtained, will, 

 it is easy to see, satisfy the general conditions; and since, if a=c, or again 

 if [26] = [a], the opposite forms (a, 6, c) and (a, —b, c), each of which 

 satisfies the general conditions, are equivalent, and therefore both belong 

 to the given class, it is clear that a form always exists satisfying the special 

 conditions proper to these cases. That only one reduced form exists in each 

 class may be proved by employing a principle due to Legendre (Theorie des 

 Nombres, vol. i. p. 77). 



"I£f=(a,b,c) be a form satisfying the general conditions for a reduced 

 form, /(I, 0) or a is the least number (other than zero) which can be repre- 

 sented byy"; andy(0, 1) or c is the least number which can be represented 

 hjf with any value of the second indeterminate different from zero." 



For, if we wish to find the least numbers that can be represented by/, it 

 will be sufficient to attribute positive values to x and y in the formula 

 f=ax^ — 2bxt/ -\- cy^, in which we suppose b positive as well as a and c. But 



f{x-l,7/)=f(x,y)-2b(x-7/)-(a~2b)x-a(x~l), 

 f(^x,y-~\)=f{x,y)-2b{y-x)-{c-2b)y-c{y-l), 



from which equations it appears that if in the formula /(a;, y) we diminish 

 by a unit that indeterminate which is not less than the other, we diminish, 

 or at least we do not increase, the value off(x,y), a conclusion which leads 

 immediately to the principle enunciated by Legendre. 



From this principle it follows that a form satisfying the general conditions 

 of reduction is the form, or one of the two opposite forms, to which we are 

 led by the process of selection above described. If, therefore, there be two 

 reduced forms in the same class, they must be two opposite forms (a, b, c) 

 and (a, — 6, c). But it is easily proved that two such opposite forms, each 

 satisfying the general conditions of reduction, cannot be equivalent, unless 



