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



and let 0=0, 1—*, 1, or —i, according as 0, 1— 1 or — i satisfies tlie 

 congruences 



^ 4- 0a = 0, mod 2, 

 ^4-ay = 0, mod 2, 



which are simultaneously resoluble, and admit of only one solution, because 

 a and y are relatively prime, while qcL—py=2. Then it will be found that 

 by the proper transformation 



^^-lr,i (^?+Sy) l 



/is transformed into an ambiguous form 0, which will be of the type (i), 

 (ii), (iii), or (iv), according as 9 = 0, 1—2, 1, or —i. It will also be seen 

 that, subject to the condition that a and y are relatively prime, there are 

 always four, and only four, solutions of the system (3), represented by the 

 formula 



There are thus four transformations included in the formula (J), two of 

 them transforming /into the same ambiguous form ^, and the other two 

 transforming /into the same form taken negatively. The four transforma- 

 tions (J), and the two ambiguous forms ^ and —<p, we shall term respect- 

 ively the transformations and the ambiguous forms appertaining to the im- 

 proper automorphic (I) . If we now form the transformations appertaining 

 to every improper automorphic of /, it can be proved (A) that these trans- 

 formations will all be diiferent, and (B) that they will include every proper 

 transformation of /into an ambiguous form. 



(A) As the four transformations appertaining to the same improper au- 

 tomorphic are evidently different, it will be sufficient to show that if (J) 

 and (J') appertain to the improper automorphics (I) and (I'), the supposi- 

 tion (J) = (J') implies (I) = (I'). From the equations 



a=a', y=y, i? + 9a=^' + 6V, gr + 9y=g,' + ay 

 (which are equivalent to the symbolic equation (J) = (J')), combined with 

 the system (3), and with a similar system containing the accented letters, 

 we find 



(e_ox=x'-x, (e-0>y=/i'-/ti, (a-a') y^=v -v, 



whence again (0 — 9') («a^+25ay + cy^)=0, by virtue of equation (2). 

 The coefficient of 9—5' is not zero, for D=6^ — «c is not a square; there- 

 fore 9 — 9'=0 ; z.e. \=X', iu=/, >' = v', or (I)=(r). 



(B) Let I f I ^ proper transformation of / into an ambiguous 



form 0; according as </» is of the type (i), (ii), (iii), or (iv), let 9=0, 

 1, or— 2; let also X=2a/3— 9a^ /i=aS+/3y-9ay, v^lyl-^y" 



then I '^'""^ I =(I) is an improper automorphic of /; for 



^2_Xv=(aS— iSy)'=l, and X«+2^5 + »'c=0, 

 because of the ambiguity of the form into which / is transformed by 



