)94 PB0CEEDING8 OF THE AMEBICAN ACADEMY. 



U Y. A i = 0,(X lt Z,) = 0, i A .A , A. 

 1 - V| '" 1IL , \.\ : 2X u i \, A, A . A ,XJ = 2X % -\ A 



* 



*-H; )( -, )■ 



Adjoined group 



do d~.dk 



- 2a *F7-»- a «znr- a 4znr' ^.xt--" 



- "4 . - 



On) 



2 a, — + (Oj + L' '/,) r- - + «3 ~ — , 

 9 11 \ 3 <'■■ a u 3 



discontinuous. Therefore, nil groups ut this type arc discontinuous; 



e. g., group 



.,5 9 9 9 9 9 



dXi 3 .'.; dari c Xj c ■' i 



T , v f (Z„ X) = 0, (A,, A' :1 ) = 0. ( A . A ) = A',. 

 • M ' L V - I ( A„ A 4 ) = A,. I A, A,) = A,. I A , A\) = 0* 



Adjoined group continuous. 



9 9 9 9 9 9 .. 



Group .- -, =- , x, 7S — , Xi «— + x t ■=— + ■= , discontinuous. 



_ „ ( ( -V,. X) = 0, (A r A ( ) = 0, (X a , X) = A, 

 ^ pe V - 1 (A,. 20 EE A„ ( X AJ = 0, (A , A,» = * 



-rr)^: 1 ) 



Adjoined group continuous. 



Parameter group discontinuous ; also group 



9 _9__ _9_ _9_ 9 



9% x 'Jx x 9r.; 9,-.: J :,: 



B. With three-parameter i aroint im* group. 



_ . ((A„ A 2 ) = (.Y, X..) = (X, XJ = 0, 



^P 61 KA-.^IE-V,.^. \ : )=j8X. (A, A')= y 



(a /^y). 

 _ /,X- ,\ A' v! -l\ ( •"-! } 



