ON THE THEORY OF NUMBERS. 143 



when f (a) is multiplied by a unit of which the Xtic character is not unity- 

 The inapplicability of the formulae of art. 30 to any general demonstration of 

 the law of reciprocity is thus apparent. The only equation of reciprocity 

 that has been elicited from them is the following : — 



V 9x A V g 2 A V 9e A V(a)A \<K a )A \<K*)A 



in which (j> (a) is a complex prime factor of a prime number p of the form 



m\ + l, and q v q 2 , q e are the e conjugate factors of a prime number q 



appertaining to the exponent/ for the modulus X. This equation, which, if 

 we adopt the generalized meaning of the symbol of reciprocity, may be writ- 

 ten more briefly thus, ($AfU) =( — £_) , was first obtained by Eisenstein, 



V 9 A Vtf>0)A 

 who inferred it from M. Kummer's investigation of the ideal prime divisors 

 of \p( a ) ( see a note addressed by Eisenstein to Jacobi, and communicated 

 by Jacobi to the Berlin Academy, in the Monatsberichte for 1850, May 30, 

 p. 189). In a later memoir (Crelle's Journal, vol. xxxix. p. 351), Eisenstein 

 proposes an ingenious method — reposing, however, on an undemonstrated 

 principle — for the discovery of the higher laws of reciprocity ; but it would 

 seem that the application of this method failed to lead him to any definite 

 result; and it is unquestionably to M. Kummer alone that we are indebted 

 for the enunciation as well as for the demonstration of the theorem. 



57. M. Kummer appears to have waited until he had developed the theory 

 of complex numbers with a certain approximation to completeness, before 

 proceeding to apply the principles he had discovered to the purpose which 

 he had in view throughout, the investigation of the law of reciprocity. He 

 succeeded in discovering the law which we have enunciated, in the year 

 184.-7, and, after verifying it by calculated tables of some extent, he commu- 

 nicated it to Dirichlet and Jacobi in January 1848, and subsequently, in 

 1850, to the Berlin Academy, in a note which also contained the demonstra- 

 tion of the complementary theorems relating to the units, and the prime 

 divisors of X. From the analogy of the cubic theorem, it was natural to 

 conjecture that the law of reciprocity would assume the simple form 



f — J = (— 2 ) f° r primes p x and p 2 reduced, by multiplication with proper 



complex units, to a form satisfying certain congruential conditions. But 

 to determine properly these conditions, i. e. to assign the true definition 

 of a primary complex prime, was no doubt the principal difficulty that M. 

 Kummer had to overcome in the discovery of his theorem. If X=3, the 

 single congruence / (a) =/(l), mod (1— a) 2 , sufficiently characterizes a 

 primary number; and since, whatever prime be represented by X, that con- 

 gruence is satisfied by one, and one only, of the numbers included in the 

 formula a. k f (a), it was probable that it ought to form one of the con- 

 gruential conditions included in the definition of a primary complex prime. 

 In determining the second condition, M. Kummer appears to have been 

 guided by a method which depends on the arithmetical properties of 



the logarithmic expansion of a complex number. If we develope log "' ^ / 



in ascending powers of ■ f ~f} - a "d represent by L ^~ the finite num- 



ber of terms which remain in this expansion after rejecting those which are 

 congruous to zero for the modulus X, we are led, after some transformations, 

 to the congruence 



