132 KANSAS UNIVERSITY SCIENCE BULLETIN. 



putting x 2 -f- y 2 ==r'-', we have r = exp<:0. But this is the polar equa- 

 tion of a logarithmic spiral about the origin cutting the axis of real 

 numbers at an angle whose cotangent is c. Hence the locus of points 

 in the complex plane which correspond to the transformations in the 

 one parameter group H(AA')c, is a logarithmic spiral about the zero 

 point cutting the axis of reals at an angle <f>, such that cot <f> = c. 



Different values of c give different spirals, each of which cor- 

 responds to a one-x^arameter subgroup of H2(AA'). The real number 

 c may assume in turn all values from - - oo to + oo, so that these 

 spirals lie infinitely close to one another. These spirals all pass 

 through the unit point. For c = o, the corresponding spiral becomes 

 the unit circle; for c = co, the spiral reduces to a straight line, the 

 axis of real numbers. 



The family of spirals for which c is positive fills the entire plane 

 and no two of them intersect except in the unit point. The same is 

 true of the family of spirals for which c is negative. But every spiral 

 of one family intersects an infinite number of times every spiral of the 

 other family; thus these spirals cover twice over the entire plane. 

 Every point in the plane not on the unit circle lies on two of these 

 spirals, from which we infer that every loxodromic transformation in 

 the group H2(AA') belongs to two distinct loxodromic one-parameter 

 subgroups. 



Every hyperbolic transformation in H2(AA'), except the involu- 

 toric transformation for which k= — 1, belongs to three one-parameter 

 subgroups ; for two spirals and the axis of real numbers pass through 

 every point for which k is real. The elliptic transformations in 

 H2(AA') belong only to the elliptic subgroup. The involutoric 

 transformation is common to the elliptic and the hyperbolic sub- 

 groups. The identical transformation is common to all subgroups, 

 and the two pseudo- transformations, for which k = o and k= oo, are 

 common to all subgroups except the elliptic. Two loxodromic sub- 

 groups for which the c's have the same sign have no transformations 

 in common other than the identical and the pseudo-transformations, 

 while two loxodromic subgroups for which the c's have opposite signs 

 have in common an infinite number of discrete transformations. 



Theorem 16. Every elliptic transformation in H-2(AA') belongs to 

 one and only one subgroup ; every loxodromic transformation in 

 H^AA') belongs to two distinct subgroups; every hyperbolic trans- 

 formation in H2(AA'), except the involutoric transformation, belongs 

 to three distinct subgroups. 



Generation of Hi(AA') from infinitesimal transformations. — 

 The same geometric representation enables us to discuss intuitively 

 the generation of finite transformations in Ho(AA') by the repetition 



