AND GENERA OE TERNARY QUADRATIC FORMS. 
293 
the sum (50), it is only necessary to consider those numbers M which are are prime to n ; 
because 
c 
=0, if M is not prime to n ; and if 
V = 8ILy X ITr X IB x lb, 
x(V)=Vxi,xn(i-l) xni(i-i) xnj(i-f) x n 
the sum (50) contains %(V) numbers M inferior to V ; let these be represented by 
x u x 2 , ... x { ; then all the numbers M, which enter into that sum, are contained in the 
%(V) linear forms xV and the sum (50) maybe decomposed into^(V) partial sums, 
of which the sum 
xv+a^- 
2 "VxV+x, 
/ — ilXi 
\ n 1 
is one. The limit of this sum is 
Qxi \ 
9 1 1 /-S 
so that the limit of the sum (50) is 
„ 1 1 _ / — nxA 
If n 2 is divisible by any prime other than the primes S, the symbols 1 — — ‘ ) are one half 
equal to +1, and one half equal to —1; in this case, therefore, the limit of the 
sum (50) is zero. But if n 2 contain no prime other than the primes S, the symbols 
are a ^ e 9 ua l to one another and to ^ ; and the limit of the sum (50) is 
1 /— I2F\ 
fx^x- 8 x(— y 
,x(v) 
V 
in accordance with the formula (51). Substituting in the expression (49) for the sum 
(50) its limiting value, we find 
iim s =^x^x|xn(i-I) x ni(i-i)xn|(i-|) 
XS-aXSS 
(52) 
In the sum 2)2 
(=f) n K) 
the summations extend to all values of com- 
posed of unequal primes c$, and to all values of n x prime to 20 ; v is any prime divisor 
of ra,, other than one of the primes eS. Thus the two summations are independent, and 
