294 CHAPTER XIII. 



Inversely, if B 2 , B 3 , B be defined by 277), the relations 271), 

 273), 275) all follow from 276). Since B^ is of period 3, 



EfEtB^EiB^EgEiEsE* (interchanging B^E 2 wi 

 E*E 2 - EfB^E* - E*E 2 E* - E* = E,E,E,B,E, 

 E 2 EE 2 Si E% E?E 2 = E 2 E 1 B 1 E 



Upon setting S 1 E 2 = E 2 S i , S 1 E^S 1 E i =E 1 S l El > we find that 



= E 2 E?E 2 S E 2 E L E 2 = E. 2 EfB^E^ = E~ ! 

 Since E 2 E V E 2 E* =E 1 E 2 E*E 2 , we get 



Since E^E 2 E?E 2 - E 3 E^E 2 E* = E 2 E E 2 E* 



= E 2 E 1 E,E 1 



=E,E,E,E, 



we find by 277) that 



7?- 1 



= A - 



upon setting E^E Z = B^= E 2 B^, E?E 2 E E 2 = E E. 2 E* and 

 applying also the equation given by taking the reciprocals of the 

 last substitutions. Using 277) and the last result, 



E~ 



In order to prove that E~ 1 BE i = B, we note that 



or 



E 



But the left member equals B 2 B 3 B. Indeed, by the earlier results, 



E-^B^E^B.B, 



Heace 



