142 REPORT 1860. 



N representing the norm of (a), B v B 2 . . . B M -i the fractions of Bernoulli, 

 and D m the value of the differential coefficient 



dm \og<p(e v ) , cT log [>(*")]* ) for V= Q. 

 dv m V hdv m 



These formulae do not in general hold for the exceptional prime numbers X, 

 which divide the numerator of one of the first /j. — 1 fractions of Bernoulli. 

 This is evident from the occurrence in them of the coefficients D m , which if 

 <j> (a) be ideal, and h be divisible by X, may acquire denominators divisible 

 by X, thus rendering the congruences nugatory. It is sufficient to have 

 determined the characteristics of the particular system of units E x (a), E 2 (a), 

 . . . E j (a), because, as that system is independent, every other unit e (a) 

 is included in the formula 



e (cc) = E 1 (a) m i E 2 (a) m 2 E M _, (a)"V-i ; 



so that x L € ( a )] ma y De found from the congruence 



k = iu-l 



xC c ( a )]= s m *X [Ejfc(a)],modA, 



k=l 



which cannot become unmeaning, except in the case of the exceptional 



primes, because if D' be the logarithmic determinant of the system of units 



Ej (a), E 3 (a), . . . E M _! (a), D and A retaining the meanings assigned to them 



D' . D' D' D 



in art. 50, it may be shown that — is prime to X, and therefore — = k y — is 

 J D ADA 



also prime to X; i.e., the denominators of the fractions m v m 2 ,. . .?«^-i are prime 



to X (see art. 42). But M. Kummer has also given a formula which assigns 



directly the characteristic of any unit e (a) whatsoever. If Aa denote the 



value of the differential coefficient sULS — £, for v=0, we have 



dv h 



X [e (a)] = A, £=!+ S~ A 2t D A _ 2 „ mod X*. 



X k=l 



56. We have already observed (see art. 39) that it is impossible to deduce 

 a proof of the highest laws of reciprocity from the formulae which pre- 

 sent themselves in the theory of the division of the circle. It is true (as we 

 shall presently see) that the formulae IV. and V. of art. 30 determine the 

 decomposition of the real prime p (supposed to be of the form k\+l) into its 

 X — 1 complex prime factors ; but it will be perceived that these complex fac- 

 tors occur, not isolated, but combined in a particular manner. From equation 

 IV. of the article cited we infer that p=\p (ac) J/ (a -1 ); let then \p (a)=/(a 1 ) 

 f(oc 2 ) . . . •/(%) ; a l5 a 2 . .cc^ being fi different roots (of which no two are re- 



ciprocals) of the equation =1 ; so that/(aj), /(a 2 ), . . ./(a M ) are one- 



half of the complex primes of which p is composed ; if e (a) be any real 

 unit, satisfying the equation e(a)=e (a -1 ), it is plain that e (aj 2 e (a 2 ) 2 . . . 

 e( % f=\, or ^(«)=±e( ai )/(« 1 )Xe(a 2 )/(a a )... X«(^)/(« M ). The 

 consideration, therefore, of the number \l> (a) cannot supply us with any de- 

 termination of the Xtic character of /(a t ) which will not equally apply to 

 /(a,)xc ( a i)« But for all values of X greater than 3, the number of real 

 complex units is, as we have seen, infinite ; and the character of any com- 

 plex prime/(a) with respect to any other complex prime evidently changes 



* The formulae of this article are taken from M. Rummer's second memoir on the com- 

 plementary theorems (Crelle, vol. lvi. p. 270). 





