Art, 5. -T. Takenouclii : 



i 2. 



Let u and fi bo two integers in k(f'). It can be shewn exactly 

 as in the case of ordinary (not complex) ninltiplication'*, that 



Eeplacing (y, (j) in this relation l:)y (//-+1. //—I). (/^ + 1. /^)> (/^- + .^'' /-«)' 

 (/^ + l + ,o, /7.) respectively, we get 



If we express these formnlae in terms of S, we obtain the 

 following recursion-formnlae for S : (n=a + ho) 



(i) when (a, h) = (0, 0) (mod. 2), 



(> S.2rAri-> = ^^'00' '^'/'-M+z' '^'/'-l-i-/' ^l ~ '^A'+l 'S^/'-l S'/'+/) » 



(ii) when (rt, /^) = (1,0) (mod. 2), 



_ S,, = S,^, Sp S,U - >^. '^.-2 s,?^, , 



() S.,aj^p = 'S/.^-i+z) Sf-i-i-p Sn — 6"'(?^ )" /SV+1 S,,^i S/r^p , 



(iii) when (a, h) = (0, 1) (mod. 2), 

 e o OS C3C' 



O.jit^l — Oft^-'jOy, — 0,14-1 »J /«_! > 

 /■> S'2/. J-/, = S,,j^.i^p >S';._i+/) *S',7 0^\?/)' 'S',,.;.! 0,._l O/.V^ , 



— jO-S'o,.-; 1^-/) ^ ^'(n)' Sy.A..2J.pSp+pS,,. — S,,+i o,,_io,r+i+/, ; 



1) Weber : Algebra III, §. 58. 



