TABER. — ON SINGULAR TRANSFORMATIONS. 595 



Adjoined group 



c/ «! ' cy «2 d a-i d tti d (to d «3 



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

 e. g., group 



9 9 9 ^ j_ R _^ _L __^ _L _^ 



5^' 9^: 9^: ''''' 9V,^ '-''' 9x,^'^''' 9x,^ 9x, 



i (X„ X,) = {X„ X,,) = (A-3, X\) = 0, 

 lype 11. <j ^^^^ ^,^ ^ ^ ^Y„ (A'2, XJ = (3 X„ (X„ X,) 



= X, + ^ X,* 



Adjoined group 



(a ^ ft). 

 e"4«_ i\ /e"*^ _ 1 



ri o 



9 r, 9 9 r> d C 



+ (rt2 /3 + fig) ^^ \- as(3 



5«3 



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

 6. g., group 



5 9 9 5, 5^„/5_^ _P_\ , _5_ 



5^/ 5^2' 5^3' ''^5^2'^"'''5^i"^^V''5^2'^'"'5:r3;+5:r,' 



^ ^^^ J (X„ X2) = (X2, X3) = (X3, Xi) = 0, 



lype 111. I ^^^,^^ ^.^^ ^ ^^.^^ ^^^.^^ -^.^^ ^ ^^.^ ^ ^^^^ (^^^ Y,) 



= Xo + A'g. 

 //*- 1\3 



*« 



a^ 



Adjoined group 



9 9 9 9 5 / , N 5 



* 5 «! 5 f/i 9 (In * d do d Us ^ d I'l 



9 a.^ ■'9 (I3 



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

 e. g., group 



9 9 9 9 9 9 9 9 



2 Xo 7c — 4" Xo ^ — + Xi ^ 4" X2 -^ ~r X3 - , - , _ , _ 

 ^ 9 Xi 9 x.^ 9 x^ 9 x^ d x^ d Xi d Xo 9 Xg 



