332 REPORT — 1861. 



which lie in the interior of the sectorial area, bounded by the positive axis ofx, 

 the arc of the hyperbola ax'' + 2ba;>/+a/''=l;^, and the straight line 



_ aV 



together with one, all, or some of the similar points on the contour of the 

 sector. The area of the sector is 



^log(T+UVP), 



whence, reasoning as before, we find 



__2VD__ /D\l . . 



log[T + UVD] \njn ^ ^ 



for the number of properly primitive forms of a positive determinant D. 

 The corresponding formulas for improperly primitive forms are obtained by 

 a precisely equivalent process. The results are, if D= — A, 



D*— I 



[2-(-i)-]«.=5vA.(5)l. (C) 



andifD= + A, 



[T', U'] denoting the least solution of the equation T^— DU^=4. 



102. Proof that each Genus contains the same number of Classes. — 

 The sixth section of the memoir also contains a demonstration of the pro- 

 position to which we have already referred (art. 98), that all the possible genera 

 actually exist, and contain an equal number of classes. This demonstration 

 is not deduced from the expression for the number of properly primitive 

 forms, but depends on an equation between two infinite series similar to the 

 equation (a) of the last article. Let ^ denote any one of the particular 

 characters proper to the determinant, and let <^ be any term in the product 

 n(l+x)' ^'^'^ ^^^ exception of the first term, which is unity, and also of 

 that particular combination of the values of x? the value of which, by the 

 condition of possibility, is also a positive unit. If X be the number of parti- 

 cular characters, 2^—2 will be the number of expressions symbolized by ^. 

 Let H and H' be the numbers of classes satisfying the conditions = 1 and 

 ip= — 1 respectively. It can be shown, as follows, that H = H'. Confining 

 ourselves, for perspicuity, to the case of forms of a negative determinant, we 

 have, by the principle of art. 87, 



2 $1 12 fi L 



\a^x'+'-2b^x7/ + c,r)' \a^x' + 2b^xi/+c,7fy 



where in the right-hand member |^J is +1 or —1, according as the num- 

 ber n satisfies the condition ^=1 or 0= — 1 ; and similarly, in the left-hand 

 member 0j.= — 1 or = + 1, according as the generic character cf the form 

 (a^, 6j., Cj^) satisfies the condition (^=1 or 0= — 1. In this equation the 



* If A = 3, we must triple the right-hand member of this equation ; as each set of repre- 

 sentations of a number by a form of determinant — 3 contains six representations, instead of 

 two. 



