138 KANSAS UNIVERSITY SCIENCE BULLETIN. 



know that k 2 = = kki. We wish to find the position of the point A' 2 . 

 The path curves of the two groups hHi(AA) and hHi(AA'i) have 

 one circle in common, viz., C, the circle through the points A, A', A'i. 

 This circle is therefore invariant under the resultant transformation 

 T 2 (AA' 2 ). Hence, A' 2 is somewhere on the circle C. The system of 

 transformations which leave A and C invariant have the group 

 property and form a two-parameter group hH 2 (AC). This group 

 contains cc 1 one-parameter hyperbolic subgroups, one for each point 

 on C, and one parabolic subgroup, Hi(A)0, which has C among its 

 path curves. 



The group ff s ( C).— There is a two-parameter group hH 2 (AO) 

 corresponding to each point on C. These ex 3 transformations all 

 leave C invariant, but they do not form a group, as we shall show. 

 There are also oo 3 elliptic transformations which leave invariant. 

 Let A be any point within C, and A' its inverse point with respect to 

 C. The one-parameter group eHi(AA') has C among its system of 

 path curves. In like manner all one-parameter groups of elliptic 

 transformations, whose invariant points are a pair of invariant points 

 with respect to C, leave C invariant, There are cc'-' such pairs of 

 points, and hence there are oo 3 elliptic transformations leaving C in- 

 variant. 



The aggregate of all transformations leaving C invariant forms a 

 three-parameter group H 3 (C). This group contains cc 3 hyperbolic, 

 cc 3 elliptic and go 2 parabolic transformations. It breaks up into sub- 

 groups as follows : 



H 8 (C) = oo 1 H 2 ( AC) + oo 2 eHi(AA') == ct? hHi(AA') + 

 orfeHifAAO + o^H'iCA), 



Evidently the group He contains cc 3 subgroups of the kind H 3 (C), 

 one for each circle in the complex plane. 



Theorem 24. All transformations of the complex plane, which 

 leave a circle C invariant, form a three-parameter group H3(C). 

 This group is composed of all hyperbolic transformations whose in- 

 variant points are on the circle, of all elliptic transformations whose 

 invariant points are a pair of inverse points with respect to the circle, 

 and of all parabolic transformations whose invariant point is on the 

 circle and whose invariant line is tangent to the circle at the invariant 

 point. 



The group ILi(iC). — There is another type of the three-parameter 

 group consisting entirely of elliptic transformations, which is closely 

 related to the group H3(C). This group of transformations of the 

 complex plane leaves invariant an imaginary circle (iC). The rela- 

 tion of this group H3(iC) to the group Hs(C) with real invariant circle 

 is as follows : 



