AND G-ENEEA OF TEENAET QUADEATIC FOEMS. 
285 
according as the inequalities (40) are satisfied, excluding all signs of equality, or 
admitting one, or two, or three such signs. Again, representing by (2y) the absolute 
value of 2y, we observe that a reduced binary form is a form ( x , y , z ) of which the coef- 
ficients satisfy the inequalities, 
(i) ;r>0, z> 0,' 
x = z - 
(ii) If x=(2y), y> 0, ’ 
If x=z, y> 0. 
(41) 
and that, by a fundamental proposition in the theory of binary forms, every class con- 
tains one and only one reduced form. Attending only to those uneven classes of deter- 
minant — 12M which are prime to 12, and comparing the inequalities (40) and (41), we 
find that the sum (38) contains (i) an unit corresponding to every pair of reduced forms 
(x, y, z), (x, —y, z) of which the coefficients satisfy none of the equalities y— 0, x=2y, 
X=z ; (ii) one-half of an unit corresponding to every reduced form of which the coeffi- 
cients satisfy one of them ; and (iii) one-fourth of an unit corresponding to a reduced 
form (if there be such a form of determinant — 12M prime to 12) of which the coeffi- 
cients satisfy the two equalities, y— 0, x—z , and of which the weight is consequently 
We thus obtain the equation 
H(12M) = X[xz —y 2 = 12M] . 
Again, attending only to those even classes of the uneven determinant — 12M which are 
prime to 12, we find that the sum (39) contains units corresponding to pairs of reduced 
forms, and half units corresponding to single reduced forms ; it also contains one-sixth 
of an unit corresponding to a reduced form (if there be such a reduced form of determi- 
nant — 12M prime to 12) of which the coefficients satisfy the three equalities x=2y, 2y=z, 
x=z, and of which the weight is consequently We therefore have the equation 
H'(OM)=2'[^-?/ 2 =12M]. 
Art. 18. According as the forms (f) are properly or improperly primitive, let 
r=S.2[^-3/ 2 =12M], 
or 
T=2.2'[#s-^=QM], 
the first sign of summation extending to all values of M not surpassing L ; so that, in 
both cases alike. 
To determine the limit of the quotient — i, when L is increased without limit, we shall 
again employ the geometric method of Gauss. For its application here the following 
preliminary lemmas are requisite, relating to the arithmetical properties of the function 
xz—y\ 
