290 V rot. ^mith. on Complex Binary Quadratic Forms. [June 16, 



ing to two different automorphics (I) and (!') are identical, an equation of 



the form (I')=T X (I) will subsist ; for if (J) and (J') are the transfor- 

 mations appertaining to (I) and (I'), since by hypothesis (J) and (J') 

 transform / into the same form, we must have an equation of the form 



(J') = (T)^x (J) ; but (J') appertains to (r),and (T)^x (J) to (T)'^^x (I) ; 



therefore (I') = (T) x (I), by what has been shown above (A). 



If then we calculate the eight ambiguous forms appertaining to the four 

 improper automorphics 



thesa eight forms will be the only ambiguous forms equivalent to /. Thus 

 every uneven ambiguous class contains eight ambiguous forms. 



Combining this result with the preceding we obtain the Theorem, 



" The number of uneven ambiguous classes is one half of the whole 

 number of assignable generic characters." 



The number of semieven and even ambiguous classes is determined by 

 the two following Theorems : — 



" When D = + 1, mod 4, there'are as many even as semieven ambiguous 

 classes." 



"When D=l, mod 2, there are as many semieven as uneven ambiguous 

 classes, or only half as many, according as there are altogether as many 

 semieven as uneven classes, or only half as many." 



To prove the first of these theorems, let J)=f^j mod 4, and let 



it is evident from the principles of the composition of forms that if (<p) is 

 a given semieven ambiguous class, the equation (S) x (^) = (1 +i) X (/) 

 is satisfied by one and only one even ambiguous class (/) ; in addition to 

 this we shall now show that, if (/) is a given even ambiguous class, the 

 same equation is satisfied by one and only one semieven ambiguous class 

 ((f) ; from which two things the truth of the theorem is manifest. First, 

 let the whole number of even classes be equal to the whole number of semi- 

 even classes*; then the equation 



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



* That if D=+l, mod 4, there are either as many semieven as even classes, or else 

 three times as many, is a theorem of M. Lipschitz (Crelle, vol. liv, p. 196), of which it is 

 "worth while to give a proof here. The number of even classes is to the number of semi- 

 even classes, as unity to the number of semieven classes satisfying the equation 



(2)X(0) = (1 + OX(/), 

 /representing any given even form. To investigate the semieven classes satisfying this 

 equation, apply to / a complete system of transformations for the modulus 1 +i, for 



1 ^'^ 1 



1 l+iO 1 



1 1 



1 0,1+'/ L 



1 .1 1, 



1 0,1 1 



