340 REPORT — 1861. 



For a negative determinant, 



4 A \m/ 2 \m'J 



the summations with respect to m and m' extending to all values prime to 

 2A, and inferior to 4A and 2A respectively. 



Dirichlet observes that when the determinant is positive, the coefficient of 



— ^ — is a logarithm of the form log (Ta + Ua V D) ; (Ta, Ua) 



log(l + U V-U) 



being one of those solutions of the equation T^— DU^=1 which are deducible 

 from the theory of the division of the circle. Thus h is in fact determined 

 as the index of the place occupied in the series of solutions of T^ — DU'^=1, 

 by an assigned trigonometrical solution. (See a note by M. Arndt in Crelle, 

 vol. Ivi. p. 100.) 



In the particular case in which the determinant is a prime of the form 

 iw + S taken negatively, an expression for the number of classes had already 

 been given by Jacobi (Crelle, vol. ix. p. 189). It would seem, from his note 

 on the division of the circle (Crelle, vol. xxx. p. 166), that the unpublished 

 method, by which his result was obtained, formed a part of tiiat theory. 



