200 



CHAPTER VIII. 



6 



, . . ., m) 



in order that it leave V absolutely invariant are seen to be the 

 special Abelian relations 1 ) 76) for ft = 1 together with the following: 



190) 



t=i 



t=l 



It follows from 114 that every set of solutions ,-y, /3^-, y f y, #/, 

 in the (^J^[2 W ] of the relations 76)^=1 leads to a special Abelian 

 substitution 



whose determinant A is unity. 2 ) 



The determinant of the coefficients of the 2m quantities x,-, tf,- in 

 the 2m equations 190) is seen to equal A. Hence ; since A =f= 0, 



It follows that S takes the form 



==1 



1/2 



=i 



i/a 



% 



the coefficients of S' being subject to the Abelian conditions 76) 

 only. The group of the substitutions S is therefore holoedrically 

 isomorphic with the special Abelian group SA(2m, 2") of the sub- 

 stitutions Z. The structure of the latter group is given in 117. 



201. Changing the notation employed in exhibiting the function Ft, 

 the canonical quadratic form for 2m indices may be written 



1) Since p = 2, we have 1 = -{- 1 in the field. 



2) For a direct proof that A =|= 0, see American Journal, vol. 21, p. 244. 



