( «^ ) 



■fib) _ _ _ _ 



.^-^ i = (p (6), if h^ « = 1, i. e. n znz h {mod. h) 



^^ ^ I = O, if /ij 7i e|e 1, i. e. »i e|e /i (moc^. 6) 



So (8) changes into 



when with the aid of 

 we eliminate h^, we get 



fib) 

 Mt,h (.) rz: J- V -7A-7-: (9) 



v=:l 



II. Let c? be ]> 1 and let b be put equal to cW, h to cZ^, so 

 that B and ^ are prime to each other. Evidently 



fi(w6 -[- A) = fi ( ^(mij -j- /z) I \ 



\ y ( or = 0, 



according to mB + -ff being prime to d or not. Hence 



""''' ^'^ =^ 2l ^nb:^ = ^ 2^ {mB^H)s ' * * ^'^^ 



where the sign ^' denotes that ??i assumes only those values, for 



which iiiB + ^ is prime to d. If m^ = ïji^ (mod. d), then it is 



evident that m^B-\-H and m^B-\-H are simultaneously prime to ^7 



or not. So those ??i distribute themselves in certain arithmetical 



progressions modulo d; i.e. among the d progressions m = 0, 1, . . ., 



d — 1 (mod. (/) m has in certain progressions, let the number be ^, 



to pass through all numbers > 0. This q is the number of those 



among the d numbers in mB-\- H,{m^=^0,l, . . . ,d — 1), which are 



{B,d)(f{d) „^, 



prime to d; this number is known to be . When the cor- 



(p{B4) 



responding values of m are denoted by m^, . . . , ifiix, . . . , mp and 



m).B-\-II is put equal to h, {X=z 1, . . . , q), then every h is prime 



to d and — on account of {B, H)=^l — to -S, so to 6 = dB and 



situated between (excl.) and b (excl.) (as 



< II -^h — nix B -j- 11^ (d—l) B + H=b—B + //^ b 

 and b itself is not prime to b). The corresponding \ alues of mB -|- H&re 



{mx -\-ld)B-\-H=lb-\-hi, 



