592 PEOCEEDINGS OF THE AMERICAN ACADEMY. 



Adjoined group continuous. 



Group Xi ^ — , Xi 7^ — — Xo-^ — , Xo -^ — , discontinuous. 



d X2 a Xi d Xo " d Xi 



Type II. (Zi, X,) = 0, {X,, X,) = X„ (X, X^) = /3 X, {(3 :\^ 0, 1)* 

 Adjoined group ~ a^ - — , — a^ (3 ^r— , Oj ^r \- cto (3 



cy «! d a2 cy «i d a^ 



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



e. g., group 



9 9,^9,9 9 



Xi ^ h P Xo -^ h a 



d Xi] d Xi d x^ d Xs d X2 



Type III. (Zi, X,) = 0, (X^, X,) = X„ {X„ X^) = X,* 



Adjoined group continuous. 



Parameter group discontinuous ; also group 



9 9 9 9 9 



9 Xi " 9 x-i 9 X3' 9 Xi 9 Xo 



Type IV. (Xi, X,) = 0, {X,, X,) = X,, (X„ X,) =X,\ X,* 

 Adjoined group 



9 9 9 , ^9 9 



— «3 ^ , — «3 ^ «3 7i , («1 + «2) 7=j 1- «2 Tj— -, 



d ai d tti d a^ d fi\ d «2 



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

 e. g., group 



c, 9 9 9 9 9 9 9 

 ^ ^3 :s 1" ^2 7^ h ^1 7^ 1- ^2 n 1" ^3 ^=r~"5 



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



Type V. (Zi, X,) = 0, {X„ X,) = X„ {X„ X^) = 0* 



6*"^- 1 



a 



■3 



Adjoined group continuous. 



