ON THE THEORY OF NUMBERS. 141 



a k for j\ [ , h being defined by the congruence hk s k', mod X. For the 



Ln a )J 



symbol ? ■ > ( so defined, the law of reciprocity still subsists, subject 



uK*)J 



however to the condition that [<£ («)]'' is primary. 



It will be seen, therefore, that the exceptional primes of art. 52 are ex* 

 eluded from M. Kuinmer's law of reciprocity, for a twofold reason : — first, 

 because if X be one of those numbers, the definition of a primary number is 

 not in general applicable ; and secondly, because, on the same supposition, 



the symbol xA_J may become unmeaning. 



55. The Theorems complementary to M. Rummers Law of Reciprocity.— — 

 The prime 1— a, and its conjugate primes, as well as the complex units, 

 are excluded from the law of reciprocity; but complementary theorems by 

 which the Xtic characters of these numbers may be determined have been 

 given by M. Kummer. For a simple unit a k , we have the formula 



N-l 



v . With regard to X, which is the norm of 1 — a, it may be 





observed that if (f> (*) be a prime factor of a real prime q appertaining, for 

 the modulus X, to any exponent/ different from unity, i.e. if q be not of the 



linear form »»X+1, the character of every real integer, and therefore of X, 



qf—\ 

 with respect tod (a) is + 1, because, if/> 1, — - — is divisible byq—l. But 



A 



whatever be the linear form of q, the characteristic of X or x(^) (for so we 



shall for brevity term the index of a in the equation — — r =a A ), is de- 



J U(«)Ja 



termined by the congruence 



X (X) = r D *» mod x > 



A 



D A being the value (for v=0) of the differential coefficient $-!—* — - 



° dv K 



if (b (a) be an actually existent number, or of ^-ii — L. if it be ideal, 



h av K 



To obtain the characteristics of the units, M. Kummer considers the system 



of independent units 



E^E^a), E^_,(«), 



defined by the formula 



—2ft -4ft -2(/t-l)ft 



E*(«)=e(a)e(« v )' e(o? 2 ) . ...e (o?*~* ) 



in which e (a) represents tne trigonometrical unit of art. 50, and y is the 

 same primitive root of X which occurs in the expression of e (a). We have 

 then, for x [E* (a 9 *)] and \(l — a! ! ), the formulae 



X [Eft (a")] == (-1/ (yf-1) f*_- D A _ 2i , mod X, 

 and x (l^)--^+^ + B 1 D,-4 2 



