262 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
Attributing to V' the value VIT 2 A', we may apply the same demonstration to the case 
in which either /or F is improperly primitive, the modulus V being supposed uneven. 
Art. 6. When neither / nor F is improperly primitive, and neither 0 nor A evenly 
even, / and F, considered separately, have no particular characters, properly so called, 
with respect to 4 or 8. But, considered jointly, they have a certain character with 
respect to 4 or 8, which we shall term their simultaneous generic character, and which 
it is important to consider here. 
If m=f{x , y , z), M=F(X, Y, Z), the representations of in by/ and of M by F are 
said to be simultaneous when x , y, z, X, Y, Z satisfy the equation 
tfX+/r+zZ=0. (8) 
This definition of simultaneous representation suffices for our immediate purpose ; we 
add, however, that if the two representations are primitive * as well as simultaneous, the 
equations 
#, y, z 
x', /, z' 
=X, Y, Z, 
( 9 ) 
X, Y, Z 
X', Y', Z' 
. • ( 10 ) 
are resoluble in integral numbers x r , y\ z\ X', Y', Z'. For, considering the first of these 
equations, we observe that [or, y , z~\ is a given solution, in relatively prime numbers, of 
the equation (8) ; let [ x y', z'] be a solution of the same equation which with [#, y, z] 
forms a fundamental set; the equation (9) is then satisfied by virtue of the characteristic 
property of the fundamental set (Philosophical Transactions, vol. cli. p. 297). Thus if 
the representations of m by /, and of M by F, are primitive and simultaneous, m is pri- 
mitively represented by a binary form of determinant — DM, which is itself primitively 
represented by/f ; and, reciprocally, M is primitively represented by a binary form of 
determinant — Am, which is itself primitively represented by F. 
In the four cases in which neither O nor A is evenly even, the simultaneous character 
of / and F is given in the Table of Art. 4. The symbol ' V P in that Table represents 
A/+1 Q'F-t-1 ' A'm+1 Q'M+l 
the unit ( — 1) 2 2 , i. e. the unit ( — 1) 2 2 , O' and A' denoting the same 
numbers as in Art. 4, and m, M being any two uneven numbers simultaneously 
represented by / and F. Thus, if Q==A = 1, mod 2, the simultaneous character ' V P 
is attributed in the Table to / and F ; i. e. the uneven numbers simultaneously repre- 
sented by / and F either all satisfy the equation 'P=l, or else all satisfy the equation 
¥=-l. 
To demonstrate these simultaneous characters we consider the four cases separately, 
and for the forms /and F we substitute forms <p and <I> equivalent to them, and satisfying 
* The representation of a number by a form is said to be primitive when the values of the indeterminates do 
not admit of any common divisor besides unity. 
t See Art. 10. 
