On the Relatively Abeliau Corpora. k 



(iv) when (a, b) = (1, 1) (mod. 2), 



The expressions of ^S; for small A-alues of a and h can easily be 

 found by direct calculation. They are 



s, = 1, s,^, = 1 +;■>, s,.„ = a-o)p, 



S, = 32;'-%^), 5n+, = (3 +;i)jr-og„ S,., = (3-/V 

 ,S,= -^jf + lOrj.p' + rj,-, 



where p stands for ^(u'). All the otlier .S' can be found by the re- 

 cursion-formulae. 



The general form of A',., is as follows'': 



'^>' = — oi^ - +c^g.p -' + 



n. - G 



when m = 



(7j being even) 



^y- — P-P - +c,g.p '-^ + +[ 



W-l »7-7 



S,.= HP - +c,g.p -' + 



when m = 1 



(7> being odd) 

 — (— 1) - if-Q'A^ p. when ;;i = 3 



) (mod. G), 



'^." = - -tj-^J - + <^i .^3P - + + [ J j/> ^, « , when m = 4 



where Ci, c,, are integers in /<;'), and ('^ J ') means Legendre- 



Jacobi's symbol for quadratic character. 



1) T. Takenouolii : Tolioku Mathematical Jonrual, Vol. VII, Xos. I, 2. 



