AND GENERA OE TERNARY QUADRATIC FORMS. 
281 
Let r be the sum of the weights of the representations of the numbers M which do not 
surpass L by the form F, and let w be the weight of f or F, so that T—w%Vi ; the equa- 
tion (29) becomes 
lim 
4 
3 
Mv) 
V 3 
(30) 
or considering in succession all the forms of (F), and observing that T = 2r, W=2w, 
iimg=*Axn(i-i) n ( i -i)n( i -i)x n [i-(^ / )y. . . (si) 
T 
which is the first determination of the limit of the quotient — r . 
Art. 15. The second determination of the limit of the quotient T -r-L^ depends on the 
following theorem : — 
“ The sum of the weights of the primitive representations by the forms (F) of a given 
number M divisible by p unequal primes, is 2^ times the weight of a genus of binary 
forms, of determinant — f2M, and properly or improperly primitive, according as the 
forms (f) are properly or improperly primitive.” 
The principles which give the demonstration of this theorem are contained in Arts. 
280-284 of the ‘ Disquisitiones Arithmeticse,’ and have been in part already employed in 
Art. 10 of this memoir. We have shown in Art. 11 that one genus and only one of binary 
forms of determinant — QM admits of primitive representation by the forms (f) of the 
ternary genus (f, F). Let <p 1? <p 2 , . . . or (<p) be a system of forms representing the classes 
of that binary genus ; these forms are properly or improperly primitive, according as the 
forms ( f) are properly or improperly primitive : let n be their number and v the sum 
of their weights ; as their weights are all equal, the weight of each of them is ^ ; so that 
each has - positive automorphics, and is transformed into any equivalent form by - 
positive substitutions. We shall first show that the sum of the weights of the primitive 
representations of the forms (<p) by the forms (f) is equal to 2 M Xi' ; and secondly, that 
the sum of the weights of the primitive representations of the numbers M by the forms 
(F) is equal to the sum of the weights of the primitive representations of the forms (<p) 
by the forms (f). 
(i) Each of the n congruences 
— A<p=(Qx—Q!yy, mod M, . . (32) 
in which Q, Q' are the numbers to be determined, is resoluble, and admits of 2^ incon- 
gruous solutions. From each such solution we deduce, by the method of Gauss employed 
in Art. 10, a ternary form f of the given genus, containing one of the forms (<p) as a part, 
and having Q, Q', M for the coefficients of 2yz, 2 xz, z 2 in its primitive contravariant. 
There are 2 M x n of these forms ( f ) ; none of them is the same as any other, and none 
of them can be transformed into any other by a substitution of the type 
2 Q 2 
