300 REPORT — 1861. ' 



with (a,b, c). If none of them be equivalent to (a, b, c), M does not admit 

 of primitive representation by (a,b,c); but if one or more of them, as 



( M, D,^, ' ~ — ), be equivalent to («, b, c), let ' ^i be the formula exhi- 



M, Oj, _i___ J ; then all the 



primitive representations of M by (a, b, c), which appertain to the value 

 iij of \/D, mod M, are contained in the Ibrmula (a, b, c)(a,, y)-=M. 



87. Determination of the number of Sets of Representations. — It appears 

 from what has preceded, that if S denote a system of representative forms of 

 determinant D (i. e. a system of forms containing one form, and only one, for 

 every class of forms of determinant D), the number of different sets of pri- 

 mitive representations of M by the forms of S is equal to the number of 

 differenb solutions of the congruence .r-^D, mod M. If, in particular, 

 M be uneven and prime to D, it is clear that M can only be represented 

 by properly primitive forms; and in this case the number of solutions of 

 the congruence ,r'-^D, mod M, i.e. the number of sets of primitive re- 

 presentations of M by the properly primitive forms contained in S, is 



expressed by either of the two formulae n|l +(—)), or ^fy)' '° which 



p and S denote respectively the prime divisors of M, and those divisors of M 



which are divisible by no square; while |— jand| — J are the quadratic 



symbols of Lagrange and Jacobi (see arts 16, 17,68,76). If /u denote the num- 

 ber of different primes dividing M, the common value of the two expressions 



II| 14-(— Hand SJ -- k is Z** or zero, according as the condition | — |= 



satisfied by every prime divisor of M, or is not satisfied by one or more 

 of them. Wiien D is ^^1, mod 4, S will certainly contain improperly 

 primitive forms ; and the unevenly even number 2M (where M is still sup- 

 posed prime to D) will admit of primitive representation only by the impro- 

 perly primitive forms contained in S (for if il denote any root of the con- 

 gruence a;^=D, mod 2M, ii will be uneven, ~ even, and the form 



|M, il, -~ ~ I will be improperly primitive). And the number of seta of 



primitive representations of 2M by these improperly primitive forms will be 

 the same as the number of sets of primitive representations of M by the pro- 

 perly primitive forms in S. 



The problem of obtaining the derived reprefentatioiis of M by (a, b, c) 

 depends on that of finding the primitive representations of a given number 

 by a given form (see art. 79). Two derived representations of M are said to 

 belong to the same set, when the greatest common divisor of the indetermi- 

 nates, which we will symbolize by w, is the same for each, and when the two 



primitive representations of — ;, from which they are derived, appertain to 



the same value of y/D, mod —„. Adopting this definition, we may enunciate 

 the theorem, "If M be an uneven number prime to D, the whole number of 



1 is 



