256 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
theory, the contravariants f and E, and the invariants 12 and A, being everywhere 
simultaneously interchangeable. 
Of definite forms we shall consider only those which are positive ; and in the case of 
such forms we shall suppose Q, as well as A, to be positive, in order that F as well as f 
may be positive. In the case of indefinite forms we shall always attribute opposite signs 
to 12 and A ; so that in this case the discriminants of f and F will he of opposite signs. 
Thus the definiteness, or indefiniteness, of a form is indicated by the signs of its invariants ; 
if, for example, p> and q are positive numbers, the forms x 2 -\-q)y 2 -\-pqz 2 ^ a?—py 2 — pqz 2 
— x 2 -\-py 2 -\-pqz 2 are respectively of the invariants [p, q\, [ — p, y], [jj, — q ] ; and their 
primitive contravariants, pqx 2 -\-qy 2 -\- z 2 , — pqx 2 -\-qy 2 -\-z 2 , pqx? — qy 2 — z 2 , are respectively 
of the invariants \_q, p\ [#, — p], [—q,p~\- 
Art. 3. A primitive form f is properly primitive when one at least of its three prin- 
cipal coefficients a, a!, a is uneven ; it is improperly primitive when those coefficients 
are all even. In an improperly primitive form, one at least of the three coefficients 
b, b', b" is uneven (or the form would not be primitive) ; if, therefore, f is improperly 
primitive, 12 is uneven and F properly primitive ; and, reciprocally, if F is improperly 
primitive, A is uneven and f properly primitive. Again, the discriminant of an impro- 
perly primitive form is always even. Whenever, therefore, 12 and A are both even, or 
both uneven, neither f nor F is improperly primitive. Primitive forms of the same 
invariants [12, A] are said to belong to the same order when they and their primitive 
contravariants are alike properly or alike improperly primitive. An order of properly 
primitive forms of the invariants [12, A] always exists, for the form 
^_|_lY + QAz 2 
is a form of that order. And we shall show hereafter that, when 12 is uneven and A 
even, there is always an improperly primitive order of forms of the invariants [12, A], 
in which f is improperly and F properly, primitive except when 12 is an uneven square, 
and }A an even or uneven square. And, reciprocally, when A is uneven and 12 even, 
there is always an improperly primitive order of forms of the invariants [12, A], in which 
f is properly and F improperly primitive, except when A is an uneven square, and ^12 
an even or uneven square. These exceptions cannot occur if the forms are indefinite. 
For example, there are two orders of forms of the invariants [1, 12]. The properly 
primitive order contains three classes, represented by the forms 
/l, 1, 12\ /l, 3, 4\ /2, 3, 3\ 
Vo, o, o/’ Vo, o, o/’ Vi, i, 1 / 
The improperly primitive order, in which the forms are improperly primitive, but their 
contravariants properly primitive, contains two classes, represented by the forms 
( 2 ’ 2 ’ 4 y ( 2 ’ 2 > 4 y 
V— i, -l, o/ Vo, o, -1/ 
