1864.] Prof. Smith on Complex Binary Quadratic Forms. 389 



appertains to (I), an improper automorphic of /. 



I ^ I . Also ^ j appertains to (I) ; for, writing p and q instead of 



2/3-0«, and 2d-h. we have | ^ ] = | i(qtly] \ ' « (of which 



a and y are relatively prime) being four numbers which satisfy the system 



It follows from (B) that, if we calculate the ambiguous forms ^ and —<p 

 appertaining to every improper automorphic of /, we shall obtain all the 

 ambiguous forms to which / is equivalent ; it remains to see how many 

 of these ambiguous forms are different from one another. If (I) 



= I [ is any given improper automorphic of /, all its similar auto* 



morphics are contained in the four formulae 



(T)2*X(I), (T)2^+ix(I), (T)2*X 1^^' ^1 I x(I), 



(T)2^+ix X(I), 



where k is any positive or negative number, and (T) 



[t^yuj representing a fundamental solution of the equation t^—Du^^l. 

 Similarly, if (J) represent the four transformations, appertaining to (I), by 

 which /passes into (p or — ^, all the proper transformations of /into or 

 —(j) are included in the formula (T)^ x (J). "We shall now show that the 

 four transformations included in the formula (T)^ x (J) appertain to the 

 improper automorphic (T)^^ x (I). Writing 



«i=(^ifc-K)a-cwA:r» I>k=(h-^'^k)P-cmy 

 y;fc=aMfca + (^fc+K)r> Qk=<^UkP + (h + ^^kh> 



H2k~ 02k— ^^2k) f^ — CU2k'' = ^^hk^ + (^2k + ^^2k)f^' 

 ^2k='^^2kfJ- + (^2fc+ 5w2fc)»'» 



we find immediately 



^1 — b, — c 



\^2icy — Xsi 



V2ki —\i2k 



ijf X (J)= I 1 ^^^t 1 . (T)2^ X (I)= 

 Also attending to the equations (2) and (3), and to the relations 



we obtain, after substitution and reduction, 



Pk^k=hky Pk72k=F2k—'^> 



^k^k=i^2k+^' qkyk=^2k^ 



f . e. (T)* X (J) appertains to (T)2^: x (I), if (J) appertains to (I). 



It follows from this result that the ambiguous forms appertaining to (I) 

 and to (T) X (I) are the same ; for /is transformed into the same forms 

 by (J) and (T) X (J) ; and conversely, if the ambiguous forms appertain- 



