294 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
But 
and 
n 1 
=n 
■ 
1 1 1 1 i 1 
■ 1 - 
1 
’¥ 
xn 
i+V+-w+-w : +-- 
_ _ 
=11 
xn- 
i i 
H h-n 
i+i 
the last sign of multiplication extending to all primes v which do not divide 2 12 A. Also 
24=n— h =n^-:xn — 
1-1 1+1+' 
so that the product 
is equal to 
or to 
because 
K) 
\ »* ) 
n i ,l 2 
n [ 1+ ( _ V E )H xn rT xII 1 J T 
1 8 2 1 v 2 
^np-ijxnp-ijxnp+^h], . 
n i n r^ n rr n i 
1 «, 2 1_ 7 2 l ~¥ v 2 
(53} 
is equal to the sum of the squares of the reciprocals of the uneven numbers, that is to — . 
8 
Substituting for the product (52) its equivalent (53) in the equation (51), we obtain the 
formula (47). 
If the forms (f) are improperly primitive, we have to employ the equation 
A'(OM)=i[2 + (-l)^ + 
7 r 
and the proof is the same as in the former case. Only, if A=2, mod 4, it is convenient, 
IF-l M 2 — 1 a 
on account of the factor 2+ ( — 1) 8 8 , separately to determine the limit T -4- L 2 for 
the numbers M which satisfy the congruences M=30, M=70, mod 8; and then to 
add the results. 
Art. 22. The weight of an order (Art. 13) is the sum of the weights of the genera 
contained in the order. The determination of this sum may in every case be effected 
by means of the formulae 
