274 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
But this equation is the equation (11) of art. 8, which is by hypothesis satisfied by the 
proposed generic character; therefore the equation (25) is also satisfied; i. e. a properly 
primitive binary form <p exists, of determinant — 12M, possessing the generic character 
which we have assigned to it. This form, multiplied by — A, is a quadratic residue cf 
M ; for the equation 
is satisfied for every prime dividing M, by virtue of the equations (23). Let, then, a 
ternary form f, of the properly primitive order, and of the invariants [12, A], be deter- 
mined, representing <p primitively. The generic character of this form is completely 
determined by the numbers m and M, which are uneven numbers simultaneously repre- 
sented by f and F; it is therefore a form of the proposed generic character. 
Of the two improperly primitive orders, it will suffice to consider that in which f is 
improperly and F properly primitive ; so that 12 is uneven and A even. Let M be a 
number prime to 212A, of the same sign as A, and satisfying the generic characters of 
F, including the congruence M= — 12, modi; also let <p be an improperly primitive 
binary form of determinant — 12M; the generic characters attributable to <p are (i) its 
characters (ii) its characters (J^j . These characters we determine, as before, by 
the equations (23) and (24). The complete generic character thus assigned to <p is pos- 
sible ; for the condition that it should be possible is 
/ — 2A\ 
Transforming ^ j by the law of reciprocity, we find 
F 2 -l sf\/V\ fi, + l Aj+lfir — 1 
(-« * x (£)(a;) =(-1) 2 ’ 2 
an equation which the proposed generic character satisfies by hypothesis (equation (12) 
Art. 8). An improperly primitive form <p of determinant — 11 M therefore actually exists, 
having the generic character which we have assigned to it ; i. e. ternary forms exist 
having the proposed generic character. 
It is evident from the demonstration that if M is of the same sign as A, prime to 2 A, 
and also (when 12 is uneven and A uneven or unevenly even) congruous to 0, mod 4, 
there is always one genus of properly primitive binary forms of determinant — 12M 
capable of primitive representation by a given genus of ternary forms of the properly 
primitive order [12, A], of which the contravariant characters coincide with the charac- 
ters of M. And similarly, if A is even, 12 uneven, M prime to A, and = — 12, mod 4, 
there is always one genus of improperly primitive binary forms of determinant — 12M 
