LINEAR GROUP WITH QUADRATIC INVARIANT. 

 Under this transformation, F takes the form 



175 



Hence 6r 6 is holoedrically isomorphic with the second orthogonal 

 group 0_i (6, p n ). Reversing equations 158), we find 



1 = 12 

 1 2| B =r 14 +r 23 , 2i 4 = 



The following substitution leaving jP 4 invariant, 



2V: 



14> 



V' r~ 



-^ T 



becomes, when expressed in the new indices 160), 



' ' li- ^ + '-'Jfc -C'-*- 1 ) 



It is always an 5> 6 , but is of the form Ql\$ if and only if T be a 

 square. It follows that 0_i(6, p n } contains a subgroup 0' of index 2, 

 which is the form taken by 4,2 when expressed in terms of the ,-. 

 The subgroup 0' may be extended to 0_i(6, p n ) by the substitution 

 5 (7 6 , the new form of T_I. 



We proceed to prove that 0' is identical with the subgroup 

 O.!_i(6, p n ) defined in 181. Expressing the orthogonal substitution 

 Of ,'4 in the indices Yy, we obtain the substitution, denoted for the 

 moment Of'?: 



is 



13 



y y 



X ^ 



24 



34 



For e==2y s -l, <J = 2yd, whence (^2 = ^, we see 

 the second compound of the substitution of determinant 



7 d 



y -d 



d y 



-tfO 



that 



s 



