594 PROCEEDINGS OP THE AMERICAN ACADEMY. 



lype lii. I ^x„ X,) = 2 X„ (X, X,) = X,, (X3, X,) = 2X, + X,.^ 



Adjoined group 



5 9 9 9^9 9 



2 «i 7=r— + («2 + 2 ag) ^ h ag 



9 cii 9 a» 9 0,3 



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

 e. g., group 



c 3 , ^ c^ 9 , 9,9 9 9 

 2 Xs h X2 ^ — , 2xi^ \- X3 ^ h 



' 9 Xi 9 x^ 9 Xi 9 x-i 9 x.^ 9 Xi 9 x^ 



j {X„ X,) = 0, (X„ X3) = 0, (X, X3) = Xi, 

 ^^^ '■ I {X„ X,) = X„ {X,, X,) = X„ iX„ X,) = 0* 





Adjoined group continuous. 



^ 5 9 9 9 , c),5,. ^. 

 Group ^ — , -^ — , Xi TT — , Xi ^ h Xo ;r 1- ^ — , discontinuous. 



9 x^ d Xi d X2 9 Xi 9X2 d x.^  



( {X„ X) = 0, (X„ X3) = 0, (X, X) = x„ 



e"^ - 1 \ /e"^-l\ 



«4 / \ «3 / 



Adjoined group continuous. 



Parameter group discontinuous ; also group 



9 9.99 9 



Xi :r — + 



9 Xi 9 Xi 9 X3' 9 X.2 "9 X.2 



B. With three-parameter involution group. 



( {X„ X,) = (X, xo = (X3, xo = 0, 



J, X4) = y A3 



(a ^ /? 4^ y). 



'^^P^ ^' "I (Ai', X]) = a xi, (X, A4) = (S X,, (A3, -^4) = 7 X3* 



A., 



/ /^"-I \ / /*' -1 \ / / ^" - 1 \ 



\ «4a y V «4/^ / \ ''47 /" 



