ON THE THEORY OF NUMBERS. 513 
D 
If the values w and w’ are concordant, tz. e, if it is possible 
' 
to find a number Q satisfying the three congruences Q7=D, mod MM’, 
Q=w, mod M, Q=v’, mod M’ (in which case the solution © of the con- 
gruence 0?==D, mod MM’, may be said to comprehend the solutions w and w' 
of the congruences w°==D, mod M, and w*=D, mod M’), the form 
1 0? —D 
MM’, Q, ww 
will belong to one and the same class (which may be termed the class 
compounded of the classes containing (a, >, c) and (a’, b', c’)) whatever two 
numbers (subject to the conditions prescribed) are taken for M and M’. 
To prove this, a few preliminary remarks are necessary. (1.) Ifthe solu- 
tions w and w’ are concordant, there is but one solution Q (incongruous 
mod MM") comprehending them. (2.) The necessary and sufficient condition 
for the concordance of w and w’ is wa’, for every prime modulus dividing 
both M and M’. (3.) If Q, w, w’ satisfy the congruence x*==D for the 
modules MM’, M, and M' respectively ; and if, besides, Q=w, Q=w', for every 
prime divisor of M and M' respectively, w and w’ are concordant, and Q is 
the solution comprehending them. (4.) The value of /D to which any 
given primitive representation (such as M=am?>+2hmn-+en*) appertains, 
may be defined by congruences, without employing the numbers p and v 
which satisfy the equation my—nu=1 (see Art. 86); in fact, we find 
am+(b+w)n=0, mod M, (6—w)m+cen=0, mod M; whence also w==—8, 
mod d, w=-+6, mod d’, if d and d@’ are common  LVIaDES of M and m 
and of M and n. 
We may suppose that the given forms (a, b, c) and (a’, b’, c’) are so prepared* 
that the representations of a and a’ by them appertain to concordant values 
of /D; 7.¢. that we can find a number B satisfying the congruences 
B?=D, mod aa’, B=), mod a, B=0', mod a’. Let state C; the forms 
a 
Mw’ ' wi? 
»W, 
will be a properly primitive form of determinant D, and 
(a, b,c), (a’, b',c’) are then equivalent to (a, B, a’ C), (a, B, a C) respectively ; 
and if X=wa' —Cyy', Y=aay'+a'a'y+2Byy', we find by actual multipli- 
cation aa'X?+2BXY + OY? = (aa? +2Bay+a'Cy’) x (av? + 2Ba'y' +aCw?). 
From this equation (which is included as a particular case in the formule 
of M. Arndt) it appears that MM’ is capable of representation by (aa’, B, C) ; 
it can also be shown (1) that this representation is primitive; (2) that 
it appertains to a value of /D, mod MM’, comprehending the values w 
and w’, to which the representations of M and M' by (a, 6, c) and (a’, 8’, c’) 
respectively appertain. (1.) If x, y, 2’, y', and X, Y are the values of the 
indeterminates in the representations of M, M’, and MM’ by (a, B, aC), 
(a', B, aC) and (aa’, B, C), the hypothesis that X and Y admit of a common 
prime divisor p is expressed by the simultaneous congruences wa’ — Cyy' = 
axy' +a'x'y+2Byy'=0, mod p. These congruences are linear in respect 
of the relatively prime numbers w’ and y's their coexistence implies, 
therefore, that p divides their determinant M; similarly it may be shown 
that p divides M’; so that w=w', mod p, because w and w’ are concordant. 
The congruences satisfied by w and w’ now give the relations av-+(B+w)y=0, 
* Tt is readily proved that a properly primitive form can represent numbers prime to 
any given number; thus a form can always be found equivalent to a given properly pri- 
mitive form, and having its first coefficient prime to a given number. ‘This transformation 
will be frequently employed in the sequel. ... In the present instance, we have only to 
substitute for the given forms any two forms respectively equivalent to them and haying 
their first coefficients relatively prime, 
1 2M 
