592 PROCEEDINGS OP 1 111- AMERICAN ACADEMY. 



Adjoined group continuous. 



Group X] r — , Xi •= x.,^ — , Xo-= — , discontinuous. 



d .'i c " 9 Xi 



Type II. (.Y I .X) = > (.Y I ..V, I = A',. « A . X,) = X t (fi * 0, 1).* 



( c" 3 - 1) (e"' p — 1 ) 

 Oa/8 



Adjoined group - a, — , — a 3 /? ■=— , a, ^ \- a, — , 



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

 e. g., group 



9 9*9,99 



d x x d Xi d x 2 c ' ■.; c?x 2 



Type III. (X,, A' 2 ) = 0, (A',. A' 3 ) = A'„ (A . A , = X,.* 



Adjoined group continuous. 



Parameter group discontinuous ; also group 



9 9,9 9 9 



<9 xi "5 a:a 9 x 3 ' 5 x^ 3 i 

 Type IV. (X„ X 2 ) = 0, (X 1? * 8 ) = X u ( A'.. A,) = A, + A 



Adjoined group 



9 9 9 . .9 9 



~ > "8 ~ "3 S » I'M T "■./ -> i '"-J •> > 



CV «l d «i & " C "i C' "j 



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

 e. g., group 



n 9 9 9 ? ? 9 9 



5 Xj p X 3 9 X, C x 2 9 X :! fC] c> x 3 



Type V. ( A,. X) = 0, ( A,. A' ;i ) = X u (A,. A.) = 0.* 



- 1 

 X= • 



Adjoined group continuous. 



