116 CHAPTER IV. 



The reciprocal of the hyperabelian substitution S is 



8' 



12 / 



Indeed, the product 88 1 replaces 2^ 



12J 



< 



{..- 



; 1 2* 



n 



1 2 J 



p n 

 2j,'2Z 



1 Silat 1 4- *2* 2? 2*) -^ 2Z 



*=i 



f 1 



Similar ly, $$ replaces |g/ by 





The relations 97) in which j > & are derived from those in 

 which j i < & by raising the latter to the power _p w . We may there- 

 fore express the hyperabelian conditions in the convenient form 



98) 



Ik 



1 (if fcj + l= even) 



(unless fc = j + 1 = even) 



( j, fc 1, . . ., 2m; j ^ fc). 

 The corresponding relations for $ 1 are found by replacing 



by respectively 



2; 



12^ 



Writing out the four sets of relations 98) according to the evenness 

 or oddness of j and Tc, and making the replacement just indicated, 

 we obtain four sets of relations for the invariance of Y by the sub- 

 stitution S~ l and therefore together equivalent to the relations 98). 

 We may combine the four sets into the single formula 



1 (if & = j + 1 = even) 

 ;_! ^ 2 , I (unless & = j + 1 = even) 



(j, fc 1, . . ., 2m; j^fc). 



hyperabelian substitution S must 



99) 



130. T/&e determinant A o/ 1 

 satisfy the relation 



100) A* n 



