272 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
— OP" ; and such a form cannot be represented by an indefinite ternary form of a posi- 
tive discriminant; f is therefore definite, and its first invariant is 4-0. When the 
given invariants O and A are of opposite signs, <p is a binary form of the positive de- 
terminant — OP"; such a form cannot be represented by a definite ternary form; f is 
therefore indefinite, and, as its invariants must be of opposite signs, in this case also its 
first invariant is -j-O. 
Also, if <p is properly primitive and P" uneven, the forms f and F' are both properly 
primitive, one of the principal coefficients of each being uneven. In this case, therefore, 
© is represented by a form of the properly primitive order [O, A]. If © is improperly 
primitive (a supposition which implies that OP"=3, mod 4), and if A is even, f is 
improperly primitive. For no properly primitive ternary form of even discriminant can 
represent primitively an improperly primitive binary form, the supposition that (p, q", 
p') is improperly primitive and p" uneven implying that the discriminant is uneven. And 
the same thing follows from the preceding analysis; for, considering the equations (18) 
as congruences for the modulus 2, we find on the supposition that <p is improperly pri- 
mitive, q= Q', mod 2, q'= Q, mod 2, and substituting in (19), p"=0, mod 2, so that/*' is 
improperly primitive. 
Art. 11. We can now assign a properly primitive form of any given invariants [O, A], 
and of any given generic character satisfying the condition of possibility. Let M be 
any number prime to 2A, of the same sign as A, and possessing all the particular 
characters (except the simultaneous character, if any) which are attributed to F in the 
given generic character ; also if Q is uneven, and A uneven or unevenly even, we shall 
suppose that M=0, mod 4. Let <p be any properly primitive, and not negative binary 
quadratic form of determinant — DM; and let m be any number prime to 212M 
which is represented by <p. By the theory of binary quadratic forms, the generic 
characters which are attributable to <p, are (i) its characters with respect to primes 
dividing M, (ii) its characters with respect to primes dividing Q, (iii) its supplementary 
characters. These last are exhibited in the following Table, 
If — OM = 
Supplementary characters. 
1, mod 4. 
None. 
3, mod 4. 
<p-i 
(-ipr 
2, mod 8. 
0 2 -i 
(-1)— 
6, mod 8. 
0-1 02-1 
(_!)-+— 
4, mod 8. 
<p-i 
(-1)- 
0, mod 8. 
0-1 0 2 -i 
(“IPA (“ 1)~ 
