1864.] Prof. Smith on Complex Binary Quadratic Forms* 285 



(5) Let us represent hy (1), (a), and (S)* respectively the principal 

 uneven, semieven, and even classes of determinant D ; i. e. the classes con- 



.2k 



taining the forms (1, 0, — D), ^1 1, — and ^2^, ?^ ^ , 



the existence of the last two classes implying the congruences D=l, mod 2, 

 D=i^^ , mod 4, respectively. Employing the formulae of M. Arndt, we 

 find (/)X (!) = (/), if (/) is any class of determinant D; (/) X (<t) 

 = (1 ^i^(J)^ if/ is derived from a semieven or even primitive ; (/) X (S) 

 =z2i(f), if / is derived from an even primitive; and, in particular, 

 (1) X (1) = (1), W X W = (l +i) W, (S) X (S) = 2z'(S). Also,if (/) and 

 {f~^) are two opposite primitive classes, (/) x (/)~^=(1), or (1 ^-^)(<7), or 

 2i(S), according as / and/~^ are uneven, semieven, or even. Hence the 

 three equations (/J X (0) = (/,), (A)X (c(>) =(!+{)(/, \ (/,) x (0)=2z'(/,), 

 in which (/J and (fj are given primitive classes, uneven in the first, semi- 

 even in the second, and even in the third, are respectively satisfied by the 



uneven, semieven, and even classes ((j>)=(f2)X-(fi) \ W= ^ — > 



(^)=^2^^ — , but by no other classes whatever. Again, let D = Am^ 



and let the forms (mjpy mq, mr), ([1 4-^]mj),mg', [1 -f i]mr), (2i'mj>,mqy 2imr) 

 represent classes derived by the multiplier m from uneven, semieven, and 

 even primitives of determinant A ; in all three forms we suppose p prime 

 to 2D ; in the second and third we suppose q uneven, and A=l, mod 2 ; 

 in the third we suppose A^P^, mod 4. The formulae of M. Arndt will 

 then establish the six equations, — 



(m, 0, — Am) X (p, mq, m^r)=(mpj mq, mr), 



^[1 + m, —m ^__l^x(p, mq, 2m>iV) = ([1 +^]mp, mq, [1 -\-i]mr), 

 ^2im, i^m, — m ^~^ ^ X (p» mq, — 4mV) = (2mp, mq, 2imr), 

 ^[ 1 + i]m, m, - m ^^"^ X ([ 1 + i']p> mq,ll-\- ^]m^r) 



= (1 ^i^mp, mq^ll + i]mr), 



^2i'm, X ([1 mq, 2i[l + i']mh^) 



= (1 +i) X (2impy mq, 2imr), 

 2im, %^m,—m \^!^ \ X {2i'p, mq, 2^mV)=2^ X {2imp, mq, 2imr). 



* It is often convenient to symbolize a class by placing within brackets a symbol 

 representing a form contained in the class ; thus (/) may be used to symbolize the 

 class contain ng the form /. 



