268 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
In the lower right-hand corner of each compartment in the Table, the number of 
possible genera contained in the order to which the compartment refers is represented 
by r ; y is the number of uneven primes dividing O, together with the number of 
uneven primes dividing A, so that if the same prime divide both O and A, it is to be 
counted twice. But it is to be observed that, when O and A are both perfect squares 
(a case which can only arise when the forms are definite), the number of possible genera 
is two-thirds of the number stated in the Table in the cases (Q), and one half in the 
cases (P). And again (as has been already stated in Art. 3), in Table B, when D is an 
uneven square, and A the double of a square, there are no possible genera ; and when 
A is an uneven square, and Q the double of an uneven square, there are none in Table C. 
Art. 9. It results from the theorem of Art. 5 that if /and F are properly primitive, 
they simultaneously and primitively represent uneven numbers prime to Cl A. We may 
therefore suppose that in /and F, a and A" are uneven and prime to OA ; we may also 
suppose that these numbers are prime to one another, because A" being prime to Cl A, 
and a being uneven, the binary form (a, b", a') is properly primitive (Art. 5), and so repre- 
sents numbers prime to its determinant. Lastly, we may assume that a and A" are 
positive. If the forms /and F are definite, a and A" are certainly positive ; if they are 
indefinite, A and O are of opposite signs ; supposing, for example, that A is positive 
and O negative, let m be any positive number primitively represented by / and M any 
number simultaneously and primitively represented by F, then M is positive as well as 
m ; otherwise mMf, which is of the type MX 2 +OY 2 -j-mOAZ 2 , would be a definite form. 
Positive numbers are therefore simultaneously and primitively represented by / and F ; 
i. e. we may suppose a and A" simultaneously positive. The complete generic character 
of/ is then determined by the characters of a and A". But 
aa! —b 2 = CIA", A'A"-B 2 =A«, 
whence it follows that 
multiplying these equations together and 
residues, 
(t)(p)=(-iP 
observing that, by the laws of quadratic 
A"- 1 
2 
we find 
(~1 F * 
(14) 
Let a and /3 retain the significations assigned to them in equation (11), Art. 8 ; trans- 
forming j an( i ^^tt) by the law of quadratic reciprocity, we find 
