ON THE THEORY OF NUMBERS. 135 



/3 representing a primitive root of the equation /3 X-1 = 1, y a primitive root 

 of the congruence y A-1 =l, mod X, and y v y 2 , y 3 , ... the least positive resi- 

 dues of y, y 2 , y 3 , . . . for the modulus \; A is the logarithmic determinant 

 (see art. 42 of this Report) of any system of fx— 1 fundamental units, and 

 D the logarithmic determinant of a particular system of independent but not 



2 u— 2 



fundamental units, e(a), e(a y ), e(a y ) e(a' ), defined by the equation 



sin — r — 2i*.nr 



/ (l- a y)(l_tt-v ) , «^-^(l-«v) , sm X .„ 



^)=V (l-a)(l-a-0 = ± 1^— =±~ ^~' lfa 



v / \ ' sin — r- 



A. 



, 11 i*— — e 



sin 



so that 



D= 



L.e(a), L.e(a y ), L.e(a y2 ), .... L.e(<r ) 



L . e(a v ), L . e(a y2 ), L . e(a v ) L . e(a v 



L . e(a y2 ), L . e(a y3 ), L . e(a y ), . . . . L . e(a ^) 



L . <«/ 2 ), L . eK" 1 ), L . e^), . . . . L . e (a^ -4 ). 



P D 



Each of the two factors — - -r — - and — , of which the value of H is com- 



(2\>-' A 



D. . 

 posed, is separately an integral number. That — is integral is a consequence 



of the relation which exists between the logarithmic determinant of a system 

 of fundamental units, and that of any system of independent units ; that P is 

 divisible by (2X)' i_1 may be rendered evident from the nature of the ex- 

 pression P itself*. The factor — , taken by itself, represents the number of 



classes that contain ideal numbers composed with the periods of two terms 

 a-t-a -1 , a 2 +a~ 2 , .... only ; or, which is the same thing, it represents the 



number of classes each of which contains the reciprocal /(a -1 ) of every ideal 



p 

 number /(a) comprehended in it ; : _ t > on the other hand, is the number 



of classes of those ideal numbers which become actual by multiplication 

 with their own reciprocalsf . The actual calculation of the factor — is ex- 

 tremely laborious, as it requires the preliminary investigation of a system of 

 fundamental units. For the cases \=5, X=7, the trigonometrical units e(a), 

 e(a y ),e(a y2 ) ...are themselves a fundamental system, so that in these two 



D P 



cases D=A, and —= + 1. The computation of the first factor _ 



A (2X)' 1 



presents somewhat less difficulty ; and M. Kummer (though not without great 



labour) has assigned its value for all primes inferior to 100. For the 



primes 3, 5, 7, 11, 13, 17, 19, that value is unity; for 23 it is 3, and then 



increases with extraordinary rapidity ; so that for 97 it already amounts to 



411322823001 =34-57 X 118982593. The asymptotic law of this increase 



is expressed by the formula 



* See the investigation in the next article. 



t See the note already cited, " on the Irregularity of Determinants," in the Monatsberichte 

 for 1853, p. 195. 



