304 



REPORT 1861. 



known theorem, the following formula for all the solutions in which T is 

 positive, 



t^+u,v/d _ /t,+u,vd Y. 



m \ m / 



k denoting any positive or negative integral number. 



From this we can can infer that it "^' 



1 Ti' ^ 

 formula (D), arising from the values Tj, Uj of T and U, all the other proper 



be the automorphic in the 



automorphics are powers of 





X 



and are included in the formula 



1,0 

 0,1 



and 



e I representing one or other of the identical transformations 



-1, 



0,-1 



If D be a negative number, the only solutions of the equation T'^ 



(except in two cases presently to be noticed) are T= + ;h, U=0. Hence 



the only proper automorphics of a form of negative determinant are the two 



-DU'=m^ 



identical transformations 



1,0 

 0, 1 



and 



-1, 

 0,-1 



The two excepted cases 



are (I) D = — 1, 7w = l; (2) D = — 3, »j = 2. In the former case we have 

 for T and U the four values +1, 0, and 0, +1 ; whence the proper auto- 

 morphics of a form of det. — 1 are the four transformations supplied by the 



— 0, — c 

 a, b 



and +1, — 1 ; whence six automorphics. 



formula 



IfD = — 3, »i=2, the solutions of T- + 3U'=4.are 



six in all, viz. +2, 0; +1, 1 

 comprised in the formula 



|(l-&), -|c 

 |a, i(l+&) 



exist for an improperly primitive form of det. —3. We may add that in each 

 of these two cases, in addition to the proper automorphics we have found, 

 there exist an equal number of improper automorphics. 



From the formula (C), compared witli the theory of representation con- 

 tained in art. 86, it follows that if (a, i, c) (a, y)- = M be any representation 

 of M by (a, b, c), all the representations of the same set are included in the 



formula I — ^s ^ — ^^ — ^-^ . For forms of a positive and 



[- 



m »* _ 



not square determinant the number of representations in each set is there- 

 fore infinite. For forms of a negative determinant the number of represen- 

 tations in each set is, in general, two ; and if [a, y] be one of them, the other 

 is [ — a, — y]. But if the determinant be — 1, or if the form be derived from 

 a form of det. — 1, the number of representations in each set is four; and if 

 the form be an improperly primitive form of det. — 3, or be derived from 

 such a form, the number of representations in each set is six. 



91. Expressioti for the Automorphics — Method of Lejeune Dirichlet.—rWe 

 have inferred the expression (D) of the automorphics of y, from the formula 

 (C) of which it is a particular case. But it is plain, from the general theory 

 of art. 81, that, when / and F are equivalent, we can conversely infer the 

 formula (C) from (D). This method has been preferred by Lejeune 

 Dirichlet, who obtains the automorphics of a primitive form/=(a, 6, c), of 



