1864.] 



Prof. Smith on Complex Binary Quadratic Forms. 



291 



is satisfied by only one semieven class {<f) ; and this class is ambiguous, 

 for the equation is satisfied by the opposite of {(f) as well as by {(f) itself ; 

 therefore (0) and its opposite are the same class, or (0) is an ambiguous 

 class. Secondly, let the number of semieven classes be three times the 

 number of even classes ; then the equation 



(s)xW = (i+0(/) 



is satisfied by three and only three different classes {(f) ; but it is also satis- 

 fied by the opposites of these classes ; therefore one of them is necessarily 

 an ambiguous class. Let that class be (^J ; the other two are defined by 

 the equations 



(1+0 (00 = M X («^o), (1+0 (<A.) = M X (0o), 



and cannot be ambiguous classes ; for by duplication we find 



(0i) X (0,) = (1+O (cr,), (0,)X(0,) = (i+O {<^r) ; 

 whereas every semieven ambiguous class produces {\-\-i)(Tf^ by its duplica- 

 tion *. 



The second theorem may be proved as follows. Let 



/=([l+^]i^, [l+^» 

 be a semieven form of determinant D ; and let 



we suppose that p is uneven. The equation {a^ X {(f) = (/) is satisfied by 

 one uneven class (^J, or by two (^o) and according as the forms 



(f>o=(Pf 9.) 22>), and <pi=(2ip, q, r), if r is uneven, or the forms 

 ^o={Py 2z>), and 0,= (2/p, [\+i'\p + q, p-^\l—i'\q-{-r')y if r is even, 

 are or are not equivalentf . If any one of the forms /, 0^, 0^ is ambiguous, 

 the others are so too ; the same thing is therefore true for the classes (/), 

 (0o), (0j. Thus the number of semieven ambiguous classes is equal to or 



and divide the resulting forms by 1 ; of the quotients, one, or three, will be semieven, 

 according as D =+l' +^' (1+0^- "^iH be found that each of these semi- 

 even forms satisfies the equation SX0 = (1+«")X/; and, conversely, every semieven 

 form ^ satisfying that equation is equivalent to one of these forms ; for, from any trans- 

 formation of (l-j-i)/into SX0, we may (by attributing to the indeterminates of S the 

 values 1, 0) deduce a transformation of modulus 1+e by which /passes into (l-j-z) ; 

 i. e., <p is equivalent to one of the forms obtained by the preceding process. It only 

 remains to show that when there are three of these forms, they constitute either one or 

 three classes, but never two. For this purpose it is sufficient to consider the three semi- 

 even forms aQ=i^-\-i, 1, — yq^^j) and o-^, obtained by the preceding process from 



the form S. These forms satisfy the equations o-oXo-o=(l-|-i)(ro, 0"iX(ri = (l+2)<r2> 

 <T^X(r^={l-\-i)(r^, (7iX(r2 = (l4-2)(To '■> ^^om which it follows that any one of the suppo- 

 sitions 0-1 = (T.^, cr2 = (ro, Co^tT^ involves the other two. 



* For the definition of the classes {<j^), (o-j), {a^ see the preceding note. 



t The forms (p^ and 0^ are obtained by applying to / a complete set of transformations 

 of modulus 1 dividing the resulting forms by and retaining only those quotients 

 which are uneven forms, 



VOL. XIII. 2S 



