NKWSON : ONE-DIMENSIONAL TRANSFORMATIONS. ]41 



infinitesimal transformation of the group. They may be generated 

 from complex infinitesimal transformations but not from real. 



Hence, we distinguish three subdivisions of h K( !i(A.V ) : those 

 transformations of the group for which k is positive and less than 1 

 constitute subdivision I ; those for which k is between 1 and go con- 

 stitute subdivision II; those for which k is negative constitute sub- 

 division III. 



Theorem 27. The group hRGi(AA') is composed of three subdi- 

 visions characterized by 0< k 1, l<k<co, k<0; subdivisions I 

 and II contain each a generating infinitesimal transformation ; the 

 transformations in subdivision III cannot be generated from either 

 infinitesimal transformation of the group. 



Properties of the group eGHi(AA'). — In an elliptic subgroup of 

 RGh the natural parameter k has the form of exp iO. The value of 

 may be restricted to the interval --«■ < < ir. The identical and in- 

 volutoric transformations of the group for which ± w, i. e , 

 k~ -1, divide the group into two subdivisions. Each subdivision 

 contains an infinitesimal transformation given by $==± 8 where 8 is 

 an infinitesimal real number ; every transformation in the group may 

 be generated from either infinitesimal transformation of the group. 



Theorem 28. The group eRGi(AA') contains two infinitesimal 

 transformations given by 6 = ± 8, each of which may generate the en- 

 tire group. 



Properties of the group pRG\(A). — The parabolic group 

 pRGi(A) has the real parameter t, and the law of combination of 

 parameters is T2 — t-f ti. Since the identical transformation is given 

 by t = o, it follows that the group has two infinitesimal transforma- 

 tions given by + 8 and — 8. Each infinitestimal transformation gen- 

 erates its corresponding portion of the group, which, therefore, is 

 composed of two subdivisions. For one subdivision t is positive ; for 

 the other, negative. 



Theorem 29. The group pRGi(A) is composed of two subdivi- 

 sions, each of which contains its generating infinitesimal transforma- 

 tion. 



TABLE OF CONTINUOUS GROUPS IN ONE DIMENSION. 



In the following table the fourteen varieties of continuous groups 

 of one-dimensional transformations are classified according to their 

 types, and the structure of each is shown. 



LOXODKOMIC GROUPS. 



First class. 



Symbol. Structure. Invariant figure. 



H2(AA') = H 2 (AA') + hIIi(AA')-f-eHi(AA'). Two points. 



J I ,i A ) = <x 2 H 2 (A A' -|- P H 2 ( A) + hHs(A) + eH 3 (A). One point. 



H c = oc 4 H 2 (A A') + oo 2 pH 2 ( A) -j- oo 2 hHs( A) + o°' 2 eH 3 ( A). No invariant figure. 



