TABER. — ON SINGULAR TRANSFORMATIONS. 593 



Parameter group discontinuous ; also group 



*1 ~ T- 



3 Xi 9 x 2 9 x x d x 3 



Type VI. (X 1} X % ) = 0, (X u X 3 ) = 0, (X, Z 8 ) = X v * 



All groups of this type are continuous. 



Type VII. (X lt X 2 ) = 0, (X 1} X 3 ) = 0, (X 2 , X 3 ) = 0* 



All groups of this type are continuous. 



Groups with four Parameters (r = 4). 

 A. Without three-parameter involution group. 



lype l - 1 (X,, Xj = 0, (X, XJ = 0, (X 3 , XJ = 0. 



e V a 2 - — 4 ctj a-, j e — v « 2 2 — 4 a, a s j 



A a = ==r -. — . 



V a 2 " - 4 «! a 3 — v <V — 4 a x a 



3 



Adjoined group continuous. 



„ 3 3 3 3 3 



Group Xj ^ — , Xi -= x 2 = — ) ^2 « — 5 #3 ~ — » discontinuous. 



c? # 2 3 #i 3 Xo a x x d x 3 



( (X u X) = 0, (X U X 3 ) = 0, (X, X 3 ) = x u 

 xype xi. | (J ^ ^ ^ p ^ ( ^ ^ ^ ^ (X ^ ^ 



A„ = 



= 08 - 1) X 3 <J3 * 1).* 



e «*_ 1\ / C ««^_ 1\ / e «40-D_ 1 



a 



4 



« 4 /3 ; v « 4 (/?-i) 



Adjoined group 



o ^ ^ 3 3 /a i\ $ o 3 



— a i ^^ — ,-a 3 ^ a 4 ~ — , «2 « «4 (p — 1) «— , «i P 



c/a! o 1 «! d a 2 cv «i a a z 9 a x 



+ « 2 ^- + « 3 (/3-l) 



3 a 2 3 a 3 



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

 e. g., group 



da 3 9 n 9 9 



x x ■= , Xi ^ [- p x 2 ■= \- 



9 x% 9 Xi 9 x 2 9 Xi ' " 9 x 2 9 x 3 



vol. xxxv. — 38 



