SLOCUM. — FINITE CONTINUOUS GROUPS. 105 



constituents of A are integral functions of the constituents of cf), and 

 therefore integral functions oi Ui . . . a^. * 



In every case for which the determinant A vanishes for certain systems 

 of values of «i . . . a^, I have found at least one group of the corre- 

 sponding structure which is discontinuous. 



I have also determined the infinitesimal transformations which generate 

 the parameter group corresponding to each structure enumerated in the 

 above mentioned table ; but since the symbols are in many cases very 

 complicated, and are of no especial interest in themselves, I have not 

 given them. 



§4. 



In this section let the variables and parameters be restricted to real 

 values. We will then consider the continuity of real groups, that is, 

 groups all of whose transformations are real. 



Let x'i =fi{xi . . . a?„, a^ . . . a,) 



(i = 1, 2 . . . n), 



in which the /"'s are analytic functions of their arguments, define an 

 r-parameter group of real transformations. Lie's chief theorem then 

 states that r real, linearly independent, infinitesimal transformations 



d_ 



(J = 1. 2 . . . r) 



in the n real variables Xi . . , x,^, generate an r-parameter real group 

 Gr if and only \i Xi . . . X^ satisfy the conditions 



r 



(;, k = \,2 . . . r), 



where the Cj^^ are real quantities independent of the X's.f 



Since the structural constants fy^,, must be real, there arc more tyj)cs 

 of structure possible for real groups than for complex groups. For ex- 

 ample, for the three-parameter structures 



(Aj, X,) = Xi, (Xi, Xg) ^ 2 X2, (Xo, Xs) = X^, 

 and 



(Xi, Xo) = — 2 Xi, (Xi, X^) = X2, (X2, Xs) = — 2 X^, 



* Taber: These rroceerlinffs, XXXV. 581. 

 t 'rransforniatioiisyruijpen, III. .%0 et seij. 



Xj r= 2, $j^ (x) 



1 d X/, 



