>96 PROCEEDINGS OF THE AMERICAN ACADEMY. 



T .... M \ i, -V. V- .V, = (X v X 1 ) = O i 



A.= 1. 



All groups of this type arc continuous. 



... TV ( (X lt X s ) = (X a , X 8 ) = (X z X x ) = 0, 



• 1,0iV ' ((-V, A',) = aA 1( (A,. I 4 ) = al a ,.\ , A\) E y A ',.« 



(" 1 y) 

 \ ". a / \ ". y J 



Adjoined group 



9 9 9 9 9.9 



C"l C", C ":, 5'/, 9 "... C ". ; 



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



6. g., group 



9 9 9 (9 9 \ , 9 9 



9% tt TO a V 1 9^ +x *9^)+ y X:i 9 x t + 9 x 4 ' 



_ ((A^X)E(X.A)E(A,.V,)En. 



lyi, ° V - 1 (X x , X 4 ) EE A",. (X s , Z 4 ) = Z 2 , (A" ; , Z 4 ) = Z a +A .' 



. « * — 



A„ 



Adjoined group 



"«- I V 



"4 / 



5 5 5 5 3 ,, , 5 , 3 



— "j T~ , — «4 « — 1 — «4 ^- — ( U n - j °i tTT + ("2 + "s) TT + " 



9 Oj' 4 5a 2 ' 4 5«., 4 5" 8 ' V": v - 8 c" 1 ", ' S 



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

 e. g., group 



9 9, 9 9 9 9 9 



TO P"x 2 TO ^S^^^^i^^Sar^^S*,' 



TV™ V ^ ( A "" ^ = (A ^' A ' :) - ( *• A "' ) - °' 



lype V - t (A". A\) = 0, (A,. AY) = 0. ( A . A,) = X,.* 



A„ = 1 . 

 All groups of this type are continuous. 



T VI << v„ xo = (X* *.) = (X 8 , X x ) = 0, 



• '"' L I ( A, . A,) EE A',. ( X,, A,) = X,, (A, A\ ) EE X s * 



(e°*- 1 



\ "' 

 Adjoined group continuous. 



