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



A 



aa' 

 "7 



an' 



__ah' 





~ H- 





a'b 









one case, either of the two real signs, and in the other case either of the 

 two unreal signs. And whichever sign we give to D, the form fk may be 



taken with either of the two real signs, if the sign of ^ is +1, and with 



either of the two unreal signs, if the sign of ^ is — 1 . In the important 



case in which c?^, d^, . . all have the same sign, we shall always suppose D to 

 have that sign, and/^, ... to be all taken with the sign +1. Adopting 

 this convention, we see that the class compounded of given classes of the 

 same determinant, or of different determinants having the same sign, is 

 defined without ambiguity. 



(3) By the general formulae of M. Arndt (Crelle, vol. Ivi. p. 69), which 

 on account of their great utility we transcribe here, we can always obtain 

 a form (A, B, C) compounded in any given manner of two forms (a, &, c) 

 and (a', h', c'), of which the determinants d snad d' are to one another as 

 two squares. 



mod A 



bn' + h'n __hb' Dnn ' 

 B — 



In these formulae D is the greatest common divisor of dm'^ and d'm^, 

 m and m' representing the greatest common divisors of a, 26, c, and a\ 2b' c' ; 



n and n' are the square roots of ^ and ^ ; /4 is the greatest common divisor 



of an' J a'uy and bn'-\-b'n. The signs of D, and n' are given, because the 

 manner of the composition is supposed to be given ; to )n we may attribute 

 any sign we please, because the forms (A, B, C) and (—A, B, — C) are 

 equivalent. 



(4) If F= (A, B, C) is compounded of two primitive forms / and/, 

 and if M is the highest power of 1 + i dividing A, B, C (so that M is 1, 



or 1 or (1 -\-iy)» the complete character of the primitive form ^ F is 

 obtained by the foUovnng rule : — 



"If w is any character common to / and /, i F will have the cha- 

 racter w= + l, orw= — 1, according as / and/' agree or differ in respect of 

 that character.'* 



In comparing the characters of / and /', it is to be observed that if w and 

 ta are two supplementary characters of /, and w x w' a supplementary cha- 

 racter of/', w X w' is to be regarded as a character common to /and/'. 



