303 iiEPOUT — 1861. 



The theory of the reduction of quadratic forms was first given by Lagrange. 

 (See his ' Recherches d'Arithmetique' in the Nouveaux Memoires de I'Aca- 

 dcniie de Berlin for 1773 ; see also his Additions to Euler's Algebra, art. 32; 

 a memoir of Euler's, " De insigni promotione scientiae numeroruni," Opusc. 

 Anal. vol. ii. p. 273, or Comment. Arith. vol. ii. p. HO; Legendre, Theorie 

 des Nombres, premiere partie, sect. viii. ; Disq. Arith. arts. 171-173 ; M. Her- 

 mite in Crelle's Journal, vol. xli. p. 193.) The method is applicable to forms 

 of a positive, as well as to those of a negative determinant ; but when the 

 determinant is positive, the reduced forms are not, in general, all non-equi- 

 valent. When the determinant is negative, it is as applicable to forms, of 

 which the coefficients are any real quantities whatever, as to those of which 

 the coefficients are integral numbers. We shall revert hereafter to the con- 

 sequences which M. Hermite has deduced from this important observation. 



We have now a complete solution of the problem of equivalence for forms 

 of a negative determinant. To decide whether two forms /j and/^ of the 

 same negative determinant are equivalent or not, we have only to investigate 

 the reduced forms 0, and (p.^ equivalent to/^ and/^ : according as f^ and (p^ 

 are or are not identical, /; and f^ are or are not equivalent; and if they are 

 equivalent, all the transformations of/^ into/^ are obtained, by compounding 

 the reducing transformation of/^, first, with the automorphics of ^^j and then 

 •with the inverse of the reducing transformation off.,. 



93. Problem of Equivalence for Forms of a Positive and not Square De- 

 terminant. — The solution of the problem of equivalence for forms of a posi- 

 tive and not square determinant occupies a considerable space in the Disq. 

 Arith. (arts. 183-196). But, as I-ejeune Dirichlet has observed, in a 

 memoir which he has devoted to this problem (" Vereinfachung der Theorie 

 der binaren quadratischen Fornien," in the Memoirs of the Academy of 

 Berlin for 1851', or in Liouville, New Series, vol. ii. p. 353), the demonstra- 

 tions relating to it maj'^ be greatly abbreviated by employing certain known 

 results of the theory of continued fractions. The following method does not 

 differ materially from that proposed by Lejeune Dirichlet ; nor indeed is it, 

 in principle, very distinct from that of Gauss, the connexion of which with 

 the theory of continued fractions he has suppressed. 



We shall suppose that the forms v.hich we consider are primitive — a 

 supposition which involves no loss of generality ; and we shall understand, in 

 what follows, by a " quadratic equation," an equation of the form 



n, + 2b^0+a,0'=O, 



in which b^- — «!„ a^ is positive, and a^, Z»„, Oj are integral numbers without any 

 common divisor. Such a quadratic equation we shall symbolize by the formula 

 [[ao,^^, ttj], and we shall regard the two quadratic equations [«o'^o'^i]' 

 [.—a„,—b^, — rtj] as different. If ^D denote the positive square root of 

 h^—Uf^a^, it is convenient to call 



a. 



-, and 



the first and second roots of [ao'^o'^il respectively; so that if we change 

 the sign of the equation throughout, we change at the same time the deno- 

 mination of the roots. Whenever therefore a root of a quadratic equation, 

 and the denomination of the root, are given, the quadratic equation itself is 

 given. Jt is readily seen that if two forms {a^^b^, aj, (A^, B„, AJ be pro- 

 perly or improperly equivalent, so that "' ^ transforms (a^, b^, a J into 



1 7) '' 



