SLOCUM. — FINITE CONTINUOUS GROUPC. 101 



where A^^ is the first minor of A relative to G^^ , and thus the ^'s are 

 integral functions of cii, lu . . . a^. Consequently, if A 4; 0, the com- 

 position of the transformation e" and an arbitrary infinitesimal transforma- 

 tion e '^ gives a transformation e""*" ^^, infinitely near the transformation 

 e"', and generated by an infinitesimal transformation. Let e" + ^''>' be 

 denoted by e% that is, let e"' = e" '" '^, where 



ai = a + 8 f y = aj Xi -f «2 ^2 + • . • + o^ X^. 



Applying the infinitesimal transformation e repeatedly, we thus obtain 

 the equations 



a, a BIfi 



a„ a, olfi a 251B 



a, an S'B a 33/S 



a,,_i 5,'^ a h5/S 



For n infinite, n^t is finite, and may be taken equal to unity ; thus 



a,, a S 



Consequently, if A does not va,nish for any system of values of a^ . . . a^, 

 in which case A is a constant,* then the composition of an arbitrary 

 transformation e" with finite pararaetei's with an arbitrary transforma- 

 tion e" = e with finite parameters, gives a transformation of the group 

 with finite parameters which is generated by an infinitesimal trans- 

 formation. 



The form of A depends only on the structural constants, and thus A is 

 the same for all groups of the same structure. Therefore, if the A cor- 

 responding to a given structure is a constant, the composition of two 

 arbitrary transformations of any group of this structure gives a transfor- 

 mation of the group with finite parameters, that is, a non-singular 

 transformation of the group, and consequently every group of this 

 structure is continuous.! 



If the A corresponding to a given structure vanishes for certain systems 

 of values of a^ . . . a,., some groups of this structure may be continuous 

 and others discontinuous. For example, the two groups 6*2 and GT, con- 

 sidered above, page 92, both have the structure (Xi , X.2) = Xi . The 



* For complex groups A is either unity, or else vanislies for certain systems of 

 values of flj . . . «,. . See the expression for A as a product on page 104. 

 t This criterion of continuity is due to Professor Taber. 



