269 



:e n{d) = ^ !i{d) = o if (y]= 0. 



dik dlD',^^ \^ J 



djm 



f 



— 27tJ»l 



in\ 



If ( " ) = 1, the equation JT B, x' =0 lias a root e * , hence 



-' 27ri;)i/ 



/=0 



Consequently 



"(:>•■ 



m 





k J ) ;— o d\k 



d\m 



for the third, second or first factor is equal to zero, according as 

 [ - I is equal to 0,1 or — 1. 



Hence 



k t 



d\m 



1 

 S, = S e~^ 2 B) e~^2 n{d) 



,n=l /=0 d\k 



dim 



= 2 Bj, ^ II ifl) 2 e d (make ??? = qcI) 



;— d\k p=\ 



I ,, 2^,H^ ^ 



= :E B, :E ii\-]:2 e ^f make -- = d' 



;-o d'\k \djp=i V d 



. ^p 27ri>(»-}-/) 



Because the sum E e '^' is equal to J' or 0, according as (?i-|- P.) 



0=1 



is divisible or not by d' , we have 



1 



5,= i^; ^ l^('^^.d' 

 y=o d'\k \dj 



d' !(»+/) 



k D'n^-, 



Since A; is squaieless, — ^ — and — — have no common factor, so 



/A;\ / /.• D'„^)\ f k \ fD'n^\ 



^^'"^ " UJ = '^ U'«+; • "^ J '■' "'"'^ '' ' U+J- '\^ cT J- ^^"'•^- 



over we have 



18 

 Proceedings Royal Acad. Amsterdam. Vol. XXI. 



