( 309 ) 



Suppose 



\k _ k' 



where k' and ij' are iiow^ prime to each other, llieii we have 



V n; k' X % Vl - e ''J = - e '' + ''^^' 



l)iU denoting hy It all integers less than h that satisfy the con- 

 gruence 



h^Ji . . . {mod. b') 

 we have evidently 



y n,/. = n',A' (D) 



h 



and also 



w^ / hk\ f h'h' 



-^ ^'i, h X lo,, \i -e V = n' k' X % VI - ^ 



hence in case /t' is not prime to h we are led to the equation 



/, — b — I / ./'^"A .k 



A^ (f{b) 



h=i 



if only we omit at the lefthand side the terms corresponding to those 

 integers h that are, multiples of //. With this limitation the equation 

 {E) applies to all valnes of /•, for if /• be prime to b, from it we 

 get back the eqnation (C). 



In this wav we obtain by putting successively /; = 1, 2, . . . ?; — 1 

 a set of b — 1 equations, from which we find in the shape of deter- 

 minants unite valnes for the b — 1 quantities l\fi. 



Actually ^ve have got more eqnations than were wanted, for we 

 may separate real and imaginary parts. 



We put 



.. -M--^ = P(..), 



so that P{.v) stands for the fractional part of the number ,6' minus 



1 

 — . Now we have generally 



1 



log (1 — ^2^'' ) = — log 4 sui^ .t .v -{- i .t P {.>;), 



Li 



hence instead of {E) we get the two ecpiations 



21 



Proceedings Royal Acad. Amsterdam. Vol. VI. 



