292 
PEOEESSOE H. J. S. SMITH ON THE OEDEES 
The last of these Tables is obtained by reciprocation from the second. 
The result in the case 12=A^1, mod 2, is given in the memoir of Eise^stein (Crelle, 
vol. xxxv. p. 128). 
Art. 21. The equation (47) may also be deduced from the theorem of Art. 15 by another 
method. We consider first and principally the case in which the forms (f) and (F) are 
both properly primitive. 
From Art. 16 we obtain the equation 
T= 
1 m<l 
. 1 V 
()<r+ < t' *4 
w M>1 
the interior sign of summation extending to every square divisor of M. Inverting the 
order of the summations, and designating by m any number prime to 20 A, we may 
write this equation in the form 
T=2^ f 2 h( OM). 
^ m = 1 M>1 
But, by a theorem of Lejeune Dirichlet, 
A(0M)=^«/0M2( 
l 
“5 
n 
the sign of summation extending to all uneven numbers prime to OM. 
T 
.ry is therefore the limit of the expression 
J_ V^ l 
2 <T+a ' 7T 
X 
m<J L 
V 
M> 1 
■OM\ 1 
n 
The limit of 
or, leaving the summation with respect to n to be effected last, of the expression 
2 <r+ ° J 7T n %n m ti \ n 
M. 
(49) 
In this expression n is uneven and prime to O ; but n is not necessarily prime to A. 
Let n—n\n 2 , n\ denoting the greatest square dividing n, so that n 2 is a product of unequal 
primes ; also let v represent any prime dividing n, other than one of the primes h ; and 
let 7i represent or 1, according as the numbers M are contained in one, two, or 
all four of the linear forms 8#-fl, 3, 5, 7 ; so that tj has the same value as in Art. 19. 
The limit of the sum 
(50) 
is zero, or 
i x (^) xn ( 1 -i) xni ( 1_ ^) xni ( 1 _ O xn ( 1- ') x i’ • (51) 
according as n 2 does or does not contain any primes other than the primes S. For, in 
