TABER. — ON SINGULAR TRANSFORMATIONS. 595 



Adjoined group 



9 9 9 9 , - 9 . 9 



— « 4 a ^ — , — aft ■= — , — a y = — , a x a = \- a 2 ft «— + a 8 y «— - 



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

 e. g., group 



9 9 9 9 9 9 ,_9_ 



9^ 9^ 9x7 3 ' aXl 9x 1 + fJX ' 2 9x 2 + yX3 9x 3 + 9x, 



_ TT l (X v X 2 ) = (X, X s ,) = (X 3 , A\) = 0, 

 lype 11. <j (Xi> A;) ^ a ^ (A . 25 XJ = ^ ^ (Xs} A;) 



= X 2 + ft X 3 * 



(a 4= £). 



"4 



A u = [ - 



e'" a 



a 4 a / \ «4 p" 

 Adjoined group 



9 9 9 a 9 9 



-a i a^- > -a i ft^—,-a i ^—~a i ft^—, c h a 



4 3 ^1 5 « 2 5«2 5 « 3 9 Hi 



1 1 



+ (a 2 + o 3 ) ^- + a 3 /3 



5 "-j 5 a 3 



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

 e. g., group 



9 9 9 9 9 ( J_ _9_\ ±_ 



tt 5^' 9x~; X3 9x~ 2 + aXl 9x 1 + l ^ V 2 9 x 2 + x * 9 xj + 9 xj 



l (X u X 2 ) = (X 2 , X) = (X 3 , xo = 0, 

 lype ill. | (x ^ ^ _ ^^ (X2j A;) _ ^ + A% ( ^ 1;) 



= x 2 + X 



Adjoined group 



9 9 9 9 9 ... 9 



9 a x 9 a x 3 a 2 3« 2 c/ «s ^ "1 



+ («, + a 3 ) ^- + a 3 ^- 



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

 e. g., group 



9 9 9 9 9 9 9 9 



2 Xo 7. + Xo ■= + X x 7. [- X 2 7: + X 3 7: , _ 5 7> , 7s • 



5 J"i 5 x 9 x x 9 x 2 9 x 3 d x x d x 2 9 x 3 



