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



representation of a binary form / of determinant D by a ternary form of 

 determinant 1 to be, that / should be a form of the principal genus ; or, 

 if D= + 1, mod 4, that /should be a form either of the principal genus, 

 or else of that genus which differs from the principal genus only in having 

 the character y= — 1, instead of 7= + !. Again, because the reduction 

 of Lagrange is applicable to complex binary forms, the reduction of Gauss* 

 is appHcable to complex ternary forms. It is thus found that the number 

 of classes of such forms of a given determinant is finite ; and in particular 

 that every form of determinant 1 is equivalent to one or other of the forms 

 —x^—y^—z^ and -^-iy^ ■\-iz^i of which the former cannot represent num- 

 bers=2, or=l+2, mod 2; and the latter cannot primitively represent 

 numbers =2, or=2(l+i), mod 4. The method of reduction itself sup- 



* \{'¥—ax^-{-a'y'^-\-a''z'^-\-^hjz-\-Wxz-\-W'xy is a ternary form of determinant A, 

 and Aj?2-j-A'y2_{.^//2^24.2B3'5'+2B'a^^+2B"a?5/its contravariant, by applying the reduc- 

 tion of Lagrange to the form ax'-\-W xy-\-a'y'^, we can render N . 2 . A" 

 (Dirichlet in Crelle's Journal, vol. xxiv. p. 348) ; and by applying the same reduction to 

 the form h!y'^-\-^yz-\-}i!'z'^, we can render N . A"^ 2 \/ N . aA. The reduction of 

 Gauss consists in the alternate application of these two reductions until we arrive at a 

 form in which we have simultaneously N . a'S. 2 \/ N . K!' , N . A"^ 2 V N . aA ; and 

 consequently N.a<4v^N . A, N . A"^ 4 . A^. If A = l, we have ]S".a^4, 



N. A"^4; whence a and A!' can only have the values 0, +1, ±_i, ±(l+«)» ±(1— 0' 

 +2, +2/; and it will be found, on an examination of the different cases that can arise, 

 that the reduction can always be continued until a and h!' are either both units, or both 

 zero. In the former case, by applying a further transformation of the type 



1, V' 

 0, 1, [I 



0, 0, 1 



the coefficients h, b', b" may be made to disappear ; and we obtain a form equivalent to 

 F, and of the type ex'^-\-e'y^+e"z^, e, e', e" representing imits of which the product is 

 — 1. In the latter case the form obtained by applying the reduction of Gauss is of the 

 type 



a'y^-\-a"z'^+2byz-{-2b'xz ; 



whence a'b"^=l, so that b' is a unit which we shall call e; and the form 

 -\-a"z^'\-2byz+2exy, by a transformation of the type 



1, 0, 

 0, 1, II 

 0, 0, 1 



is changed into one of the four forms e^y'^-\-2exz, e'^y^+z'^+^exz^ e^y^-\-iz^+2eXi^, 

 e^y^+{l-j-i)z^+2e.xz; of which the first two by the transformations 



-li, 0, e-i 



ei, ei, e 

 0, -i, -1 



0, 0, 

 H, 0, 

 0, i, 



are changed into the form 

 0, 



'X'^—y'^—z^ ; the last two by the transformations 



6, 



1, 0, 

 0, -1 



e-i, 6-1, 6-1(1-2) 

 — e, — e, ei 



0, -1, 



are changed into x^+iy^-\-iz^ (See Disq. Arith. art. 272-274.) 



