284 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
to the sum of the weights of all the primitive representations of the forms (< p) by the 
forms (f); because every primitive representation of a form (<p) by a form (f) is simul- 
taneous with one and only one primitive representation of M by a form (F). 
Combining the conclusions (i) and (ii), we obtain the result enunciated at the beginning 
of this article. 
Art. 16. Let a represent the number of uneven primes dividing Q, counting those 
which also divide A ; let <7'= — 1, when — 1, mod 4* ; when 0=0, mod 8 ; 
and (f=0 in all other cases. Let also A(OM) and A'(OM) be the weights of the properly 
and improperly primitive orders of binary forms of determinant — OM ; then x v 
— or according as the forms (f) are properly or improperly primitive. 
If A 2 is any square divisor of M, the sum of the weights of those representations of M 
M 
by the forms (F), which are derived from primitive representations of -g by the same 
'(3 
forms, is 
A 2 ) 
C 2 
Therefore the sum of the weights of all the representa- 
tions of M by the forms (F) is 
S -K^r 
or — , the signs of summation extend- 
Oa+cr' 5 O 
ing to every square divisor of M. Or, if we represent by H(QM) the sum of the weights 
of those uneven binary classes of determinant — OM which are prime to O, and by 
H'(OM) the sum of the weights of those even classes of determinant — OM which are 
prime to O, the sum of the weights of all the representations of M by the forms (F) is 
H(riM) H'(OM 
2 <r+<r' 5 Or £ <r+a' ’ 
according as the forms (f) are properly or improperly primitive. 
Art. 17. We now consider the sums 
%[xz-tf= OM], (38) 
2'O-y^OM] (39) 
In both the sign of summation extends to every solution in integral numbers of the 
equation 
xz— y 2 =OM, 
in which the greatest common divisor of x, y, z is prime to O, and in which x, y, z satisfy 
the inequalities 
#>0, y> 0, z>0,l 
• (40) 
x = z 1 
But, in the first sum, one at least of the two numbers x and z is uneven ; in the second, 
x and z are even, and y is uneven. The symbol [xz— ^ 2 =OM] is 1, or or or 
* If this congruence is satisfied by any one number of the series M, it is satisfied by every number of that 
senes. 
