296 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
contain primes not dividing 12 ( and A,, by factors placed outside the sign of summation; 
we thus find 
R'=n(i-i) 2 {ni[(^) + (^)i]xni[(S) + (^)i]}, 
where only primes <a, which divide Oj and primes which divide A, occur after the sign 
of summation. We then substitute for each factor containing a> l or ^ a factor of the 
form 
i{M^) = 0 +[-i+ {=?) i]}= (^f) h 
or 
*{[ 1+ (x) £] + [“ 1 + (tt) 0 = (t 1 ) l 
outside the sign of summation ; and observing that by the law of reciprocity 
-A N 
we find 
R' = 
Q, + I A] + l A, 2 — 1 Q, 2 -l 
= —( — 1) 2 2 X« 8 X/3 8 
n t +i Aj_ 
• 1 ) 2 • ! 
OA. 
Xa 
Xj3~ 
xn 1-3). 
As an example of the application of these formulae, let us consider the properly primi- 
tive order in the case in which A = l, mod 2, 12=4, mod 8. We may determine sepa- 
A/+1 
rarely the weights of those genera for which ( — 1) 2 = — 1, and of those for which 
(— 1) 2 = + l. In a genus of the former kind the characters 
©• G 
(56) 
may have any assigued values because the condition of possibility is 
fi, + l A. + l 
Therefore the sum of the weights of these genera is X^Xk or X R, because 
£ =^. But in a genus of the latter kind the characters (56), or some of them, are 
subject to the condition 
(£)(|)=(-fi^; • ( B ?) 
we have therefore to consider a sum of which the general term is the same as that of R, 
but into which only those terms are admitted which are formed with values of and 
g-j satisfying the condition (57). This sum is expressed by the formula 
/ Oi + l A! + l 
a(r + (_1)— — R( 
