1867.] 



Ternary Quadratic Forms. 



389 



stration) the forniulse which assign the weight of a given genus or order 

 of forms of an uneven discriminant. These formuljE are demonstrated in 

 the present Paper, and, with them, the corresponding formulae relating to 

 the cases in which the discriminant is even. The demonstration is obtained 

 by a method similar to that employed by Gauss and Dirichlet for the de- 

 termination of the number of binary classes of a given determinant. The 

 sum of the weights of the representations, by a system of forms represent- 

 ing the classes of any proposed genus, of all the numbers contained in 

 certain arithmetical progressions, and not surpassing a given number, is in 

 a finite ratio to the sesquiplicate power of the given number when that 

 number is supposed to increase without limit. Of this limiting ratio, two 

 distinct determinations are obtained ; of which the first contains, as a 

 factor, the weight of the proposed genus ; and an expression for that 

 weight is obtained by a comparison of the two determinations. Of these 

 determinations, the first is obtained immediately by an elementary applica- 

 tion of the integral calculus ; the second depends on an arithmetical 

 theorem, which is deduced in the Paper from the analysis employed by 

 Gauss in arts. 279-2S4 of the ' Disquisitiones Arithmeticse,' and which may 

 be expressed as follows : — 



" The sum of the weights of the representations of a given number 

 (contained in one of certain Arithmetical Progressions) by a system of 

 forms representing the classes of a ternary genus, is equal to the weight of 

 a genus of binary forms, of which the determinant is the product, taken 

 negatively, of the given number by the second invariant of the ternary 

 forms." 



By this proposition the determination of the limiting ratio is made to 

 depend on an approximate determination of the weight (or, which is here 

 the same thing, the number) of the binary classes of certain series of nega- 

 tive determinants. Two methods are given in the Paper for effecting this 

 approximate determination. The first method presupposes Lagrange's 

 definition of a reduced form., and depends ultimately on the evaluation of 

 the definite integral 



/// (xz-^y') dx dy ds=^~, 



of which the limits are given by the inequalities 



^^•^0, y^O, z>0, .v^z, 2y<i\ 



The second method employs the expression obtained by Lejeune Dirichlet 

 in the form of an infinite series for the number of binary classes of a given 

 determinant, and is thus independent of the definition of a reduced form. 

 The same result is obtained by both methods ; but the second is more 

 easily extended to the case of quadratic forms containing more than three 

 indeterminates. 



VOL. XV. 



