TABER. — ON SINGULAR TRANSFORMATIONS. 693 



Parameter group discoutinuous ; also group 



9 d 9,9 



^IT^ • + 



9 Xi 9 x^ 9 Xi 9 X3 

 Type VI. (Xx, X) = 0, (X\, X,) = 0, (X„ X,) = Xi* 



All groups of this type are continuous. 



Type VII. (Xx, X,) = 0, (X„ X,) = 0, (X„ X,) = 0* 



All groups of this type are continuous. 



Groups with four Parameters (r = 4). 

 A. Without three-parameter involution group. 



( (X„ X,) = X„ (Xx, X,) = 2 X,, {X,, Z3) 

 ^yP^ ■"• 1 (Xx, X,) = 0, {X,, X,) = 0, (X3, X,) = 



g V «2^ — 4 fli as _ 1 g — V 32^ — 4 a-, flj _ 2^ 



V a^^ — 4 Ox 03 ~ V "s"^ — 4 ax «: 



3 



Adjoined group continuous. 



Group Xi ^ — , Xi 7;; X2 :^ — , x^ ^ — , X3 ^ — , discontinuous. 



^9x2 9xi 9x2 9 xi 9x3 



( (Xx, X) = 0, (Xu X,) = 0, (X2, X3) = Xx, 

 lype 11. j ^^^^ ^^^ ^ ^ ^^^ ^^^^ -^^^ ^ ^^^ (^^^ ^^) 



= (;3-l)X3(/3 4^1). 



^«-V «, JV «4/3 JV «4(/3-l)/ 

 Adjoined group 



o5 5 P 9 /oix^ /jc' 



9^,'~'^'9a, '''9a.: '^'9a, "* ^^ ^ 9 a^ ^' ^ 9 a. 



5 f'.2 5 f/s 



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

 e. g., group 



9 9 9 J_M R -L^A^ 



9^: 9^; '''9x: ^^5xx"^^^'5x,'^5x; 



VOL. XXXV. — 38 



