NEWSON : ONE-DIMENSIONAL TRANSFORMATIONS. 133 



of an infinitesimal transformation. Every spiral passes through the 

 unit point, and, corresponding to the two points on a spiral adjacenl 

 to the unit point, we have two infinitesimal transformations belonging 

 to a one-parameter group. These are given by k exp(c- i)S0 and 

 k =exp — (c-fi)^- The indentical transformation divides the 

 group Hi(AA')c into two portions, each of which contains an in- 

 finitesimal transformation. Every finite transformation in each 

 portion of Hi(AA)c can be generated by the repetition of the cor- 

 responding infinitesimal transformation. 



Since every loxodromic transformation in H-2(AA') belongs to two 

 distinct subgroups, it follows that every such loxodromic transforma- 

 tion can be generated from either of two distinct infinitesimal trans- 

 formations. In the elliptic group, for which the spiral reduces to the 

 unit circle, every finite transformation can be generated from either 

 elliptic infinitesimal transformation. Every hyperbolic transforma- 

 tion for which k is positive can be generated from three infinitesimal 

 transformations, while every hyperbolic transformation in Hi(AA') 

 for which k is negative, except the involutoric transformation, can 

 be generated from two distinct loxodromic infinitesimal transforma- 

 tions, but not from either hyperbolic infinitesimal transformation. 



Theorem 17. Every hyperbolic transformation in H-i(AA'), for 

 which k is positive, can be generated from three distinct infinitesimal 

 transformations belonging to H>(AA') ; every other finite transforma- 

 tion in H>(AA') can be generated from two distinct infinitesimal 

 transformations of the group. 



Three-parameter suhgroups of H\(A). — The four-parameter group 

 H4(A) breaks up, as we already know, into go 2 two parameter sub- 

 groups H-2(AA'). We shall now show that the transformations in 

 H4(A) may be distributed into oc 1 three-parameter subgroups. The 

 law of combination of parameters k within the group H-i(A) is ex- 

 pressed (Theorem 12) by kki==ka. Written in another form, this is 



exp(co + i)0,>=expj (o'+i )0+ (ci + i)0 (. 



If ci = c, we have exp(c2 + i)#> = exp(c + i)(0+0i); whence C2 = c and 

 $., -= -(- Oi. Hence we see that if we choose from each of the two- 

 parameter groups, H2(AA'), in H4(A), the one-parameter group 

 characterized by a certain constant value of c, the aggregate of the 

 transformations contained in these one-parameter groups forms a 

 three-parameter group H y (A)c. It is clear that there is one such 

 group in IMA) for every real value of c. 



In particular, all elliptic transformations in H4(A) form a group 

 eH:3(A) ; likewise, all hyperbolic transformations in Hi(A) form a 

 group hH:3(A). 



