276 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
taneously with the representations of Mj and M 2 by and F 2 *. Then <p, and <p 2 are 
properly primitive ; their generic characters with respect to uneven primes dividing O 
will coincide, because 
(sM9-(i)-(?> 
their supplementary characters will also coincide ; for the same supplementary characters 
are attributable to $ 1 and <p 2 , and these supplementary characters are determined for <p, , 
in accordance with the supplementary characters oif \ , or the simultaneous character of 
f x and F n and for <p 2 in accordance with characters which are the same with these ; 
lastly, if jO. is any prime dividing both M, and M 2 , the characters of <p, and <p 2 with respect 
to [A will also coincide ; for 
(?)=(?)=©• 
The remaining characters of <p, and <p 2 (i. e. their characters with respect to primes 
dividing only one of the two numbers M, and M 2 ), being characters with respect to 
different primes, cannot be incompatible. The complete generic characters of <p, and <p 2 
are therefore compatible, and are satisfied by the numbers contained in certain arith- 
metical progressions. Each of these progressions contains (by the theorem of Lejeune 
Dirichlet) an infinite number of positive and negative primes. Let p be one of these 
primes of the same sign as Cl, and not dividing 20A ; p will satisfy the generic cha- 
racters both of <p, and <p 2 , and will be represented by some form of determinant -OM„ 
and of the same genus as <p n and by some form of determinant — OM 2 , and of the same 
genus as <p 2 . Therefore, by the lemma of this article,^, will be primitively represented 
by <p„ and pd\ by <p 2 , 0, and d 2 denoting numbers prime to 2f2A. Let <!>„ <h 2 be two 
properly primitive binary forms represented by F 1? F 2 , simultaneously with the repre- 
sentations of p6\, pQl, byf x ,f 2 . The determinants of d>„ <F 2 are — Apff\, —Ap&l; and 
it will be found (as in the case of the forms <p,, <p 2 ) that the generic characters of <!>,, <l> 2 
are compatible ; and that a prime P of the same sign as A, and not dividing 20A, is 
assignable, such that P©,, P0 2 are primitively represented by d> 2 respectively, 
0, and 0 2 denoting numbers prime to 20A. Thus the numbers p&\, P0, are simulta- 
neously and primitively represented by/, and F^; the numbers p&\, P0 2 are simulta- 
neously and primitively represented by f 2 and F 2 . We may therefore suppose that \J/, is 
a form equivalent to f, in which a x —p6\, A''=P0^, and that \p 2 is a form equivalent to 
* If 
H=F(a'/3"-a"/3', a"/3-a/3", aj3'-a'/3), 
and if f is transformed into a binary form <p by the substitution 
x=a.x +(3y, 
y=a'x+p'y, 
z=a"x-\-(3"y, 
the representations of M by F, and of <p by/, are said to be simultaneous, or to appertain to one another (Gauss, 
Disq. Arith. Art. 280). 
