ultimately ; and 



ON THE THEORY OF NUMBERS. 355 



if 21 = dy, mod I'. But [-) = (y), because be =E — 1, mod 8, and 



© = (A) = (^) = (fs)' because yy - -1 ' mod8; so that 



;ely; and 



^., j[-«x^m)] ,. 



(1 — a-)* 4> 8A (?'<r) 



the sign of multiplication extending to every divisor § of », and to every term 

 I of a complete system of residues of its conjugate divisor <5'. Observing that 



every factor of the numerator, in which ( -_ ]= — 1, is finite, and that every 



factor, in which { _) = -j-l, is evanescent, and is of the same dimensions as 



J 



e~ a or e « ,<r , according as %>-8', or o<<)', we see that, in order to obtain a finite 



value for lim JK ' — , we must take for X twice the sum of those divisors of n 



which satisfy simultaneously the equation ( -) = -f 1, and the inequality 



£< V h, so that we shall have 



\=4[d>(n)— ¥(n) + *'(»)— *'(»)]• 

 Further, if j; is any eighth root of unity, it will be found that, when 

 n= — 1, mod 8, / ( -, vrj )=/(m, v), whence f(- )=/(#), or f{x) contains 



only powers of x having exponents divisible by 8. Consequently f{a?) is 

 divisible by 



( cr "_l)i[*f")-'*( n )+*'(«)--'«''('»)] j 



and the quotient is prime to m s — 1. 



Eepresenting, as we may now do, any root of the equation - /v ; — = 



by </>(uj), we find that w must satisfy the equation 



1 _/2\ / rw + 16£\ 



for one system at least of values of y, y, and Tc ; that is (Table A, iii. art. 

 125), w satisfies a quadratic equation of the form 



e+d(o yw + 16Jc 

 a-\-bdi y 



or 



16aJc - Y 'c+(ay' -dy' + 16Jcb)w + byw i ~0, 



2 b2 



