SLOCUM. — FINITE CONTINUOUS GROUPS. 105 



constituents of A are integral functions of the constituents of <£, and 

 therefore integral functions of a t . . . a r . * 



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

 of values of a x . . «,., 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 =f i (x l . . . x n , ffli . . . Oj) 



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



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

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

 states that r real, linearly independent, infinitesimal transformations 



X,EE| ft ^(*)A 



(J = 1,2 • • . r) 



in the n real variables x x . . . x n , generate an /'-parameter real group 

 G r if and only if X x . . . X satisfy the conditions 



r 



(Xj, X k ) E5 ^ s c jks X s 

 (;, k = 1, 2 . . . r), 



where the c jks are real quantities independent of the X's.f 



Since the structural constants c jks must be real, there are more types 

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

 ample, for the three-parameter structures 



(Xi, X 2 ) = Xi, (Xi, X 3 ) EE 2 X 2 ,-\Xo, X 3 ) EE -X3' 

 and 



(-Xi, X) = - 2 X t , (X u X s ) = X, (X, X 3 ) =-2X s , 



* Taber: These Proceedings, XXXV. 581. 

 t Transformationsgruppen, III. 360 et seq. 



