ON* THE THEORY OP NUMBERS. 



301 



sets of representations of M (and if D^l, mod 4, of 2M) by a system of re- 

 presentative forms of determinant D is 2| — j; d denoting any divisor of D." 



We may add that, as before, M will be represented only by properly primi- 

 tive forms ; and, when D^l, mod 4, 2M only by improperly primitive 

 forms*. 



88. Reduction of the Problem of Transformation to that of Equivalence.— • 

 It has been shown in art. 80, that the general problem, " Given two forms of 

 unequal determinants, to decide whether one of them contains the other, 

 and if so, to find all the transformations of the containing into the contained 

 form," can be reduced to the simpler problem of the equivalence of forms. 

 For the sake of clearness we shall here point out how the first of the two 

 general methods of that article is to be applied to quadratic forms. If of 

 two forms/ and F the former contain the latter, the determinant of F is a 

 multiple of that of/ by a square number, viz. by the square of the modu- 

 lus of transformation. Let the determinant of/ be D, and that of F, De" ; 

 also let m and ^i be any two conjugate divisors of e, so that m^=e. Then 

 every transformation of which the modulus is e may be expressed in one way, 



m,k 

 0, IX 



and one only, by the formula 

 numbers 0, 1, 2, 3,. . . .?« — I, and 



y, I 



in which k denotes one of the 



a,/3 

 y, c 



is any unit-transformation whatever. 



formula 



(of which the number is equal to the sum of the divisors of c), 



If, therefore, we apply to the form /all the transformations included in the 

 m,k 



we shall obtain a series of forms <p,, (p.„. ... of determinant De'. If none of 

 these forms be equivalent to F, F is certainly not contained in /; but if one 



more of them, for example, <p, arising from the transformation ' 



a,/3 



or 



is equivalent to F, let 



7' 







'J" 



represent indefinitely any transformation of (p 



the formula 



If we take in succession for f every form in 



into F ; then / passes into F by any one of the transformations included in 



m, k ^ ci,ii 



0, jj. y,C 



the series <p^, (p^,. . . . which is equivalent to F, it is readily seen that the 

 transformations of/ into F, which are thus obtained, are all diflTerent, and 

 that they include all possible transformations of/ into F. 



We have supposed the number e to be positive, i.e. we have supposed 

 that/ contains F properly. To decide whether/ contains F improperly, we 

 have only to examine whether any of the forms f^, f^.... be improperly 

 equivalent to F ; and if any one of them be so, to combine the transforma- 

 tion of/ into it, with its (improper) transformations into F. 



89. Problem of Equivalence.-^lt remains to speak of the problem of 

 equivalence. Of the three parts of which this problem consists, viz. (1) to 

 decide whether two given forms are equivalent or not, (2) if they are, to 



* The theorems of this article will not be found in the Disq. Arith. If, in their expression, 

 we transform the symbols ( ^ j, ( -r) by the law of reciprocity, we obtain results which co- 

 incide with those given by Lejeune Duichlet in bis memoir, " Recherches sur I'application 

 etc.," sect. 7 (Crelle, vol. xxi. p. 1-6). 



