188 CHAPTER VH. 



Taking the reciprocal of the substitution of 6r which has 



&+& + !, 



we obtain in G a substitution S in which a\^ -f- | x =(= 1. Then 6r 

 contains the product 



where S a = S~C t C 2 S is of period two and has the form 



1 \ / m 1 



KJ^J + ^ fifmilm J 2^-2 I Ci^ -f 



/ 



\ 

 I 



$! is not the identity since $ would then be commutative with QCg 

 and would therefore break up into the product of 



{ - a 11 ii + % 2 i > ?2 = 2i Ei + "22 2 Ki + li = !) 

 by a substitution on | 3 , . . ., | w . 



We readily obtain the transformed $ a - of S a by an orthogonal 

 substitution 0, in which i,j<k: 



where by 145) 

 173) a 



We have . = 



yy 2 = 1 (I = 1 if 7c < w; ^ = ft if & = m). 

 0^(80). But S' = SO has the coefficients 



*j EE a sl (s = 1, . . ., m; s 4= ^;^ ^)- 



If ?i+a|i+ Aa^i=|=0, we can find solutions in the GrF[p n ~\ 



of 173), which make a'u = 0. We suppose a a =j= 0, the trans- 



formation of S a being unnecessary if a be already zero. Eliminat- 



ing a from 173) and 



174) aaa+/3^i 



we find the single condition on /3 and 



175) 



-f 



If 2 i+a|i=0, so that ^i=^=0 and a A1 =j= 0, this equation deter- 

 mines /3 ? when y is assigned any value =f= in the field. Then 174) 

 determines a in the field. But, if afi + |i be =j= 0, we multiply it 

 into 175), which then takes the form 



