NEWSON : ONE-DIMKNSIONAL TRANSFORMATION'S. 137 



A'. There is a different system of spirals for each value of o. Fo] 

 c==o and c = go this system of spirals reduces, respectively, to ellipl Lo 

 and hyperbolic systems of circles. 



Theorem 22. The path curves of a one-parameter loxodromic gron 1 1 

 Hi(AA')c form a system of double spirals about the invariant points 

 A and A'; the path curves of eHi(AA') and hHi(AA') are, re- 

 spectively, elliptic and hyperbolic systems of circles about and through 

 A and A. 



Path c>rrvesofH\{A)6. — The path curves of the one-parameter 

 parabolic group Hi(A)0 may be determined in the following manner: 

 The normal form of a parabolic transformation is 



dn— ^x + t- (41) 



Suppose that z is a fixed point and zi a movable point whose locus we 

 wish to find. Equation (41) may be written : 



zi — z = — t (z — A) (zi — A). 

 Putting 



(zi — z) =R'exp?>, — t = /3exp^(#+7r), (z — A)=expi'i/', 



(zi — A) = R exp l<j>, 

 this becomes 



R' exp i<f>' = p exp i(0 + tt) . r exp i\p . R exp i<$>. 

 Whence we have R' == p rR and <£' = = + it -f <£ -f if/. (42) 



Since t/s and it are constants, we have <f>' — <£ = const. Thus we see- 

 that the locus of zi is a circle passing through A and z, two fixed 

 points. 



If now z be given different positions in the plane, we have a system 

 of circles all passing through A. The angle $ varies with the posi- 

 tion of z, but 6 is a constant ; consequently, all circles of the system 

 are tangent at A, and 6 is the angle which the common tangent line 

 at A makes with the axis of real numbers. Hence the path curves of 

 the group Hi(A)0 consist of a parabolic system of circles tangent to 

 each other at A and to the line through A which makes with the axis 

 of reals an angle 0. 



Theorem 23. The path curves of the one-parameter group Hi(A)0 

 consist of a parabolic system of circles through A and having in 

 common the lineal element A0. 



§ 8. Other Subgroups of Ho. 



The (/roup ////■>( A (J)— The developments of §6 enable us to find 

 some other subgroups of great importance in He. Let us consider 

 the hyperbolic group hH 3 (A) ; it contains, as we know, oo 2 subgroups 

 hHi(AA'). The resultant of any two transformations, hT(AA') and 

 hTi(AA'i), in hH 3 (A) is a hyperbolic transformation hT,(AA',>). We 



