596 PROCEEDINGS OF THE AMERICAN ACADEMY. 



All groups of this type are continuous. 



Type IV. f (^'i' ^2) = (^2, ^3) = (^^^1) = 0, ^ ^ 



(«4=r) 



\ Cda J \ Ui y J 



Adjoined group 



a a d d B d 



discontinuous. Therefore, all groups of this type are discontinuous ; 



e- g., group 



9 9 9 (9 9 \ 9 9 



-^ — J Tij— , ^ — ? "i^i 0^ + ^20 — +y2^3o — +n — • 



( (Xi, X2) = (X2, X3) = (A'3, A'O = 0, 



Adjoined group 



5 9 9 9 9 , , , ^ 9 , 9 



c/«i 5a2 5ao 5a.g c/('i 5^/2 i? «3 



discontinuous. Therefore, all groups of this type are discontinuous; 

 e. g., group 



9 9, 9 9 9 9 9 



d Xi d X2 9X3 d x^ d Xi d X2 d x^ 



, f (x„ X2) = (X2, X3) = (X3, xo ~ 0, 



/P^ • 1 (Xi, X4) = 0, (X, X4) ^ 0, (X3, X4) = X2.* 



A„ = 1. 



All groups of this type are continuous. 



f (Xi, X2) = (X2, X3) = (X3, XO = 0, 



# 



A„ 



Adjoined group continuous. 



