140 REPORT — 1860. 



54. M. Kummers Law of Reciprocity. — We can now enunciate M. Rum- 

 mer's law of reciprocity. It appears, from the last article, or it may be 

 proved immediately by dividing the N— 1 residues of <p (a) into X groups 



of - """ .. terms, after the following scheme, 

 A 



A 



and proceeding as in art. 33 of this Report, that if >//(*) be any actual coni- 



N-l 



plex number prime to </>(a), \p(a) A is congruous for the modulus $(a) to 

 a certain power a k of a. This power of a may be denoted by the symbol 



N-l 



zA5JL • so that we have the congruence [il/fa)! * = zSzL =a fc , 



mod <b (a). The symbol v ; { which we may term the Xtic character 



L<p WJ a _ 

 of \p (a) with regard to <p (#), is evidently of the same nature as the corre- 

 sponding symbols with which we have already met in the quadratic, cubic, 

 and biquadratic theories, and admits of an extension of meaning similar to 

 that of which they are susceptible. Availing himself of this symbol, M. 



Kummer has expressed his law of reciprocity by the formula xifii = 



U> (a)J a 



Wr\ ><t>(l) and ^ (°0 denoting real or ideal primes. But, to interpret 

 UP (a)J a 



this equation rightly, it is important to attend to the following observations. 

 (1.) When \p (<t) and <p (a) are both actual numbers, the formula supposes 

 that they are both primary prime numbers. The prime. 1— <x is therefore 

 excluded. 



(2.) The definition that we have given of the symbol -?-,-£ becomes 



LVWJk 

 unmeaning when ty (#) is ideal, because no signification can be assigned to 



an ideal number which presents itself, not as a modulus or divisor, but as a 

 residue. Let, therefore, It denote the index of the lowest power of <\> (a) 

 which is an actual number ; i. e., let h be the exponent to which the class of 

 <p (a) appertains; and let C^(^)] A represent the actually existing primary 

 complex number which contains the factor <p (a) h times, but contains no 



other prime factor ; then the symbol \ ' ■ has by the preceding defini- 

 tion a perfectly definite meaning. Let then ? y{ =a k ' ; we mav define 



Up («) J a 



the value of the symbol , ) [ by means of the equation ■ • \ ' = 



U(»)Ja DM") J 



t *** ^ \ =cc k ', which, if h be prime to A, always gives a determinate value 

 \p{«.)J 



