NEWSON : ONE-DIMBNSIONAL TRANSFORMATIONS. L39 



The elliptic system of coaxial circles which form the path curves of 

 a one-parameter elliptic group contains imaginary as well as real 

 circles. Let us select from the path curves of the group eHi(AA') an 

 imaginary circle (iC), with its center at a real point O and radius 

 equal to iR. The center O is on the line AA', between A and A', and 

 the points A and A' are a pair of inverse points with respect to iC; 

 whence we have OA . OA' = - R 2 . In the case of a real circle C, we 

 have OA . OA' R 2 . All transformations in the group eHi(AA') 

 leave invariant the circle iO and also cc 1 other circles. 



There are oo 2 pairs of points in the plane which are inverse points 

 with respect to iC. These all satisfy the relation OA . OA' = — R 2 . 

 Each of these pairs of points are the invariant points of a one- 

 parameter group of elliptic transformations, and iC is one of the path 

 curves of each of these groups. Thus we see that there are ac : > trans- 

 formations which leave iC invariant ; these form a three-parameter 

 group H 3 (iC), which contains only elliptic transformations. Evi- 

 dently the group H 6 contains cc" subgroups H 3 (iC), one for each 

 imaginary circle in the complex plane. 



Theorem 25. There are cc 3 transformations of the complex plane 

 which leave invariant any given imaginary circle iC ; these form a 

 group Hs(iC). This group is composed entirely of elliptic trans- 

 formations whose invariant points are a pair of inverse points with re- 

 spect to (iC). 



No other subgroups of H*. — A circular transformation transforms 

 points into points and circles into circles. We have considered all 

 possible groups which leave one or two points invariant; a trans- 

 formation leaving invariant more than two points is identical. "\\ e 

 have also considered all possible groups of transformations leaving a 

 circle invariant. If there be a continuous group characterized by the 

 invariance of some curve other than a circle, such a curve must 

 be the path curve of a one-parameter group. The only other path 

 curve besides the circle is the double spiral of Holzm filler. This has 

 two singular points, and is invariant only under those transformations 

 whose invariant points are these two singular points; hence, there is 

 only one one-parameter group leaving invariant such a double spiral. 

 These considerations indicate that there are no other subgroups of Hg. 



§9. The Real Subgroup of H 6 . 



/ill; a special case of JHz(C). — In the geometry of the complex 

 plane a straight line is regarded as a special case of a circle, and since 

 every circle in the plane is the invariant circle of a three-parameter 

 group H3(C), it follows that every line in the plane is also invariant 

 under a group H a (L), isomorphic with H 3 (C). One of these groups 

 RH;s(L) leaves invariant the axis of real numbers in the complex 



