296 Prof. Smith on Complex Binary Quadratic Forms, [June 16, 



D=2«, or 0, mod 4, one of the two forms of determinant 1, and of the 

 type 



which are deducible by the method of Gauss from those two values, satisfies 

 the condition p"=0, or = l, mod 2, while the other satisfies the condi- 

 tion ^"=e, or 1 + 2, mod 2 ; that is, /is capable of primitive representation 

 by both the forms —x^—y'^—z^ and s^-\-iy^-^is^. 



The preceding theory supplies a solution of the problem, " Given a form 

 of the principal genus of forms of determinant D, to investigate a form 

 from the duplication of which it arises/' Let /=(«, ^, o) be the given 

 form, and let us suppose (as we may do) that a and c are uneven. "When 

 D=f, or 1+2, mod 2, let 



a,/3 

 a", /3" 



be a prime matrix (of which the determinants are A, B, C) transform- 

 ing 3^-\-iy'^-\-iz^ into («, —h, c) ; and let represent the binary form 

 (C— e'B, A, iC— B) ; then the matrix 



//3' + 2/3",ft /3, -^(/3'-^/3")^ /^x 

 Va' + zV, a, a, -ia}^)J ^ ^ 



transforms / into ^ X ^* ; and is a prime matrix, for its determinants 

 C — 2B, 2 A, and 2*0— B are not simultaneously divisible by any uneven 

 prime (because A, B, and C are relatively prime), and are not simul- 



g'^ =0, or 1, mod 2, we see that ^"=0, 1, or=«, 1+z, mod 2, according as — = 0, 1, 



or = 2, l-j-?', mod 2, But ^^^^^^^^^ ^^-^^"-^^==(1 — «')/?; + ^. ; which is con- 

 gruous to 1+2, mod 2, if D=0, mod. 4, and to i, mod 2, if 'D=2i, mod 4, since ^ is 

 evidently uneven in either case. From this it appears that if ■■ =0, 1, mod 2, 



then =% 1 +2> mod 2 ; that is, in one of the two forms f+p"2!^-{-2qy^+ 



2q'xz, p"=0, or 1, mod 2, and in the other p"=i, or 1+z, mod 2. 

 * This assertion may be verified by means of the identity 



+ (?o^'3 +i^o?3 - -i'i?2) (i'o^^'+jPi^y'+i'2^V+i>3^>') 



+ -I'o^a) (s'o^^'+^i^y +2'2^>+?3^y )^ 



in which we have to replace the quantities 



PoPlI>2P3 

 ?0 9l $2 ?3 



by the elements of the matrix (Z). 



