820 REPORT — 1861. 



to the primitive representations of M by q' — Dp" (the derived representations, 

 corresponding to the different square divisors of M, are to be treated sepa- 

 rately by the same method), we see tiiat, since p and M are prime, q is of 

 the form Mr + i2/j, where r and il are two new indeterminates of whicii the 

 latter may be supposed < [| M]. On substituting this value for q, it will 



£2,2 J) 



appear that N=: — — — is necessarily integral, i. e. that flt is one of the roots of 



the congruence Si^ — D;^0, mod M ; and the equation will assume the form 

 Nj9" + 2i2jor+Mr^ = I, in which every admissible value of ii is to be employed 

 in succession. The development of either root of the equation N + 2iiy + 

 M0^=O will give all the values of p and r which satisfy the equation 

 N/j^ + 2iljo»'+M?'^^l, because 1 is the nmiiimun value which the form 

 (N, ii, M) can assume. (See the Additions, paragraph 2, and especially 

 arts. 33-35.) Or again, if we apply the transformation of art. 92 to the 

 form (N, O, M), we obtain an equation of the typeRi'-4-2Qa;' ?/'-|-Ry-=l, 

 in which Q"— PR=D, and P< VD; whence, ifa;"=lV + Qy, we finally 

 deduce a'"'"— D?/''=P, all the solutions of wiiich (see art. 96, viii.) are neces- 

 sarily given by the development of VD in a continued fraction. Applying 

 either of these methods (the latter is not given in the Memoir, but only in the 

 Additions to Euler's Algebra) to every equation of the form Np--^2Clp--\- 

 M»-^=l which can be deduced from the equation q'^ — T)p^=M, or from the 

 equations of similar form obtained by replacing M by the quotient which 

 it leaves when divided by any one of its square divisors, we obtain a finite 

 number of formulae of the type 



_a T+(iU+y _g' T-H/3'U + y' 



[T, U] denoting any solution of the equation T^ — DU^=1. These for- 

 mulae are fractional ; but by attending to the principle of art. 96, v., we can 

 ascertain for each pair of formulas whether they contain any integral values 

 or not, and if they do contain any, we can substitute for the single pair of 

 fractional formulae a finite number of pairs not containing any fraction. 



The form in which the solution of this problem has been exhibited by 

 jrauss is remarkable for its elegance. Let 

 a, b, d 



b,c, e =/\j and S representing the greatest common divisor of b^—ac, 

 d,e,f 



cd-be, ae-bd, let ^=D', ^= A', ^-^=^=», "^I^=q, then putting D'a;= 



Co 



X+p, B'ij=Y+q, we find aX- + 2bXY +cY^=B'A.'. If [X„,Y„J denote 

 indefinitely any representation of D'A' by (a, b, c), we have only to separate 

 (by Lagrange's method) those values of X„, Y„ which satisfy the congruences 

 X.„+p^O,Y"'+q^O, mod D', from those which do not, and we shall obtain 

 a finite number of formulae, exhibiting the complete solution required. 



98. Distribution of Classes into Orders and Genera. — The classes of 

 forms of any given positive or negative determinant D, are divided by 

 Gauss into Orders, and the classes belonging to each order into Genera. 

 Two classes, represented by the forms (a, b, c), («', b', c'), belong to the same 

 order, when the greatest common divisors of a, b, c and a, 25, c are respec- 

 tively equal to those of a\ b', c', and of a', 2b', c'. Thus the properly primi- 

 tive classes form an order by themselves ; and the improperly primitive 

 classes form another order. To obtain the subdivision of orders into genera, 

 it is only necessary to consider the primitive classes ; because we can deduce 

 the subdivision of a derived order of classes from the subdivision of the 



1 



