TABER. — ON SINGULAR TRANSFORMATIONS. 591 



§8. 



For certain types of structure every root of the characteristic equation 

 of the matrix cf)„ is zero irrespective of the values of the parameters a. 

 In this case A„ = 1, and the group is continuous. 



For two groups of the same structure, but such that A„ ± 1, one may 

 be continuous and the other discontinuous. As remarked above, if the 

 adjoined is discontinuous, group G and every group of the same structure 

 is discontinuous. 



I give below a table exhibiting the result of an investigation by my 

 pupil, Mr. S. E. Slocum, of the continuity of all types of groups with 

 either two, three, or four parameters. It will be seen that for every 

 type of structure for which A„ ^ 1, there is at least one group which is 

 discontinuous. Mr. Slocum has found that every real group is continu- 

 ous which possesses any one of the several structures distinguished in the 

 table by an asterisk ; and that the real group which possesses the struct- 

 ure marked in the table thus (f) is discontinuous. Also that every real 

 group of the structure {Xi, X,) = 0, (Xi, Xg) = — X,, (X.,, X3) = — Xi, 

 is discontinuous. For this group 



_ (e"' ^^^ _ 1) (e - «3 i/~i _ 1) 



« i 



[In what follows (Xj, X^) denotes Xj X^ — X^ Xj.^ 



Groups with two Parameters (r = 2). 



e"- — 1 



Type I. (X„X,) = X,.^- A„ 



Adjoined group continuous. 



Parameter group discontinuous ; also group 



d 9 9 



9 X2 9 X2, 9 Xi 



Type II. (Xi,X2.) = 0.* A„=l. 

 All groups of this type are continuous. 



Groups with three Parameters (r = 3). 

 Type I. (X„ X,) = X„ (X„ X,) = 2 X„ (X„ X3) = X,.t 



y' Uo^ — 4 «! as ~ V "2^ — 4 ai 03 



