LINEAR GROUP WITH QUADRATIC INVARIANT. 159 



170. The conditions that the substitution 



Si ei-* (-l,...,m) 



shall leave F^ invariant are the following: 



143) *l+ !, + ..-+ !,_!, 



= m) 



144) tf^ffi* H ----- h m-l^m-lt + P<Xmj<Xmk= 



0'; & = !, .., m; 

 It follows readily that the inverse of S is 



m-l 



The determinant of S l is seen to equal the determinant D of S. 

 Hence D 2 =l ; being the determinant of S~ 1 S = I. 



Writing the relations 143) and 144) for the substitution S~ l , 

 we obtain the following relations, which are evidently together 

 equivalent to the set 143) and 144): 



146) KjitXkl-i ----- h >w-l *-!+- <Xjm<Xkm= 



(j, fc 1, - ., w; j + *)- 



171. The substitutions leaving .F u invariant were proven to have 

 determinant + 1. Among them occur substitutions of determinant 1, as 



Or. - -fc, ;-& (j = l,...,;j + *)- 



The group 0^ (m, p n ) of all linear substitutions leaving F^ invariant 

 has therefore a subgroup of index two 0^ (m, p n ) composed of all 

 linear m-ary substitutions in the 6r-F[p w ] of determinant unity which 

 leave F^ invariant. The latter substitutions will be called orthogonal^) 

 For ft = 1, we have the first orthogonal group O^m, p n )] for m even 

 and [i = v, we have the second orthogonal group O v (m, p n ). 



1) This unusual restriction of the term orthogonal to substitutions of 

 determinant -j- 1 is done in the interest of the later terminology and notation. 

 We will be concerned with such substitutions alone. If it became necessary to 

 consider substitutions of determinant 1 which leave Fp invariant, they might 

 be designated extended (erweiterte) orthogonal substitutions and the group On(m,p n ) 

 designated the extended orthogonal group. 



