Ou the Eelatively Abelian Corpora. 



and we may put'^ 





if )!i, he odd, 

 it" ))i he even, 



Avhere ÂV is a rational integral function of \^-^[/f) whose higliest term 

 is 



fA-'itt) 



or 



— ctWÖ - y 



according as m is odd or even. The sign of (/'a is thus completely 

 defined. And it can be seen that, if we expand ^v ii^ a,n ascending 

 power series of u, the first term is always equal to 



ÎÙ 



m ■ 1 



Now, 1)y (1), 





Since the expression on the left-hand side vanishes only when 



fxu = +11, (mod. oj, (1)'), 



it follows that the numerator on the riglit-hand side can differ from 

 (/'.u^i v^e-i only by a constant factor. Putting u=0, we find that this 

 constant factor is equal to —1. TJierefore 



If we assume that 



n = a + ho, 



a and /> ]>eing rational integers, we get 



s-.™)-i'» = -|^. 



when (a, h) = (0, 0) 



^ _ n^6y-.SWi>Vi ^ when (aJj) = i\,0) 



Sß. 



when (a- b) = (0, 1), (1, 1) 



{mod, t2). 



1) Cf. Weber: Lehrbuch der Alo-ebra, Vol. Ill, §. 58 and ?, ]52. 



