334 



REPORT 1865. 



the transformations, of any square order n 2 , and of the type 



n,0 i 



r. are the only 



0, n\ J 



transformations which do not alter the value of to. For if w= c ^~ c —, an( j 



a + ba 

 to=£l, we have bu> 2 -\-(a— d)u>— c=0. But, by hypothesis, to is not the root 

 of any quadratic equation having integral coefficients ; neither is to rational ; 

 therefore the three numbers b, a—d, and c are all zero, and the matrix 



a, b 

 e,d 



is of the type 



n, 

 0,n 



But if w is the root of a quadratic equation hav- 



ing integral coefficients, an infinite number of transformations, other than those 



included in the formula 



n,0 

 0,« 



, can be assigned, which do not alter the value 



of to. Let A -f 2B w + do 2 = be the equation satisfied by w ; and let AC — B 2 = A ; 

 then A is different from zero and positive ; also A and C are of the same sign, 



and may be supposed to be positive, so that to 



_ — B+tVA, 



; lastly, let 0=1, 



or =2, according as (A, B, C) is properly or improperly primitive. Let n be 

 any number such that 2 n admits of representation by (1, 0, A) ; and let 

 a, t be the values of the inde terminates in any such representation ; then the 

 transformation 



rA 1 

 '0 



(<x-rB) 



of order n will not alter the value of to, because or— — rA +Q— rB )^ 



<7 + rB + 7Ca> 



and will have for the reciprocal of its multiplier 



The transformations derived from different values of n, or from different re- 

 presentations of the same value, are all different ; and every transformation 

 of order n which does not alter the value of to, is derived from some repre- 

 sentation of d 2 n by (1, 0, A) ; so that the transformations and representations 

 correspond to one another one by one, and are equal in number. It will be 

 observed that the multiplier corresponding to any of these transformations is 

 a complex factor (composed with V —A) of the number expressing the order 

 of the transformation ; so that the transformation is equivalent to a complex 

 multiplication of the argument. And the Theta functions containing to do, or 

 do not, admit of complex multiplication, according as to is or is not a quadratic 

 surd. 



If we consider the values of to contained in Theta functions admitting of 

 multiplication with i */ A, we see that these values are infinite in number ; 

 each form of determinant —A supplying one. But the values of f 8 (io), cor- 

 responding to these values, are finite in number, being six times as many as 

 the classes of forms of det.— A; provided that in the enumeration of- the 

 classes a class of det. — 1, or a class derived from a class of det. — 1, is counted 

 as | instead of 1 ; and an improperly primitive class of det. — 3, or a class 

 derived from such a class, is counted as i instead of 1. For it appears from 

 the Table (A) that the values of <j> s (w) corresponding to two equivalent forms, 

 are equal or not, according as the transformation, by which one form passes 



