AND GENERA QE TERNARY QUADRATIC EORMS. 
297 
so that the sum of the weights of the genera of the latter kind is 
Adding the two sums together, and substituting for R and R' their values, we find for 
the weight of the proposed order the expression 
HA 
16 
n l— 
I2A 
32 
according as 0,A, is not or is prime to the greatest common divisor of 12 and A. 
If, in general, we represent the weight of any proposed order of the invariants [12, A] 
by the expression 
DA 
xzxn(i-l). 
the following Table (with which we shall conclude this memoir) will assign the value 
of the coefficient Z, and will thus serve to ascertain the weight of the order*. The 
determinations contained in it have been obtained by the method just described ; x is 
or 0, according as D, 1 A 1 is or is not prime to the greatest common divisor of O 
and A ; I 15 I 2 are the exponents of the highest powers of 2 dividing 12 and A respec- 
tively. 
(A). — (f) and (F) properly primitive. 
I 1 = 0. 
Ij even. 
I, uneven. 
I 2 =0. 
H2-*) 
i(2-X) 
i 
I 2 even. 
i(2-x) 
1(2-7.) 
1 
2 
I 2 uneven. 
X 
2 
A 
A 
(B). — ( f ) improperly, (F) properly primitive. 
© 
A 
1 
I 2 even. 
A(2-x) 
I 2 uneven. 
i (i-x) 
* For the case of uneven invariants, the result has been given by Eisexstein (Crelle, vol. xxxv. p. 123) ; 
there is, however, a slight discrepancy. According to Eisensteix, a is not zero, when the greatest common 
divisor of A and ft is a square ; according to the definition in the test, A is always zero, escept when the expo- 
nent of every uneven prime common to A and ft is even both in A and ft. For the invariants (p 2 , p 3 ) the 
weight assigned by the formula of Eisexsteix is |_^2 — ^1 — Vj, p denoting an uneven prime ; a result 
which can hardly be right, because the weight of each genus separately is =0, mod p 3 . 
