136 KANSAS UNIVERSITY SCIENCE BULLETIN. 



Theorem 21. A linear fractional transformation of the complex 

 plane is a direct circular transformation. 



Path curves of the group H\{AA')c. — The effect of the successive 

 transformations of the group Hi(AA)c upon a point P of the complex 

 plane is to move it to new positions, Pi, P>, P3, etc., which lie on the 

 path curve of the point P. By a transformation T(AA')c every 

 point of the plane is moved along its path curve. The character and 

 properties of these path curves are now to be determined. 



Starting with normal form, 



z, — A' / 1 • \ s, z — A' 



= exp(c -f- \)H- 



Zl -A — l-V- 1 -/u Z _ A > 

 for a given value of 6, z is transformed into zi ; for another value of 0, 

 z is transformed into z?, etc. The point z may thus be regarded as a 

 fixed point, and ^~ A = C, a constant. Let z' denote the moving 

 point zi, z->, etc. We thus get : 



z' — A' = Cexp(c+i)0z'— A. 



Let z'i — A = r' exp i4>, C = p exp i\j/, and z' - - A = r exp i <$> ; we then 

 have 



r' exp i<p' == p exp iif/ . exp(c + i)0 . r exp i<i>. 

 Whence, r' = pexp id r and <£'==^+ $ + $. (40) 



These two equations enable us to determine the path curves of the 

 group Hi(AA')c. 



In the first place, let c = o; we then have r'=pr, where p is a 

 constant. The locus of a point z, which moves so that the ratio of its 

 distances from two fixed points A and A' is a constant, is a circle 

 having A and A' for a pair of inverse points. Hence, the path curves 

 of the elliptic group eHi(AA') form an elliptic system of coaxial cir- 

 cles having A and A' for vanishing points. 



In the second place, let exp(c + i)0 = pexp id be real; whence 

 = 0. We now have <£' — <£ = ^, where ty is a constant angle. The 

 locus of a point z, which moves so that the angle AZA' is constant, 

 is a circle passing through A and A'. Hence, the path curves of the 

 hyperbolic group hHi(AA') form a hyperbolic system of coaxial 

 circles intersecting in A and A'. 



In the third place, let us consider the most general case where c in 

 exp(c + i)0 is any real constant. In the equation r' == p exp cO r, the 

 quantity p exp cO is not constant, but increases as increases from o 

 to 2tt, from 2tt to 4tt, etc. Thus the locus of z is not a circle returning 

 into itself, but a spiral about both A and A'. It is known as the double 

 spiral of Holzmiiller. The equation, <£' — $ — ^-f-0, shows that the 

 spiral cuts the hyperbolic system of circles through A and A' at a con- 

 stant angle \p. Hence the path curves of the loxodromic group 

 Hi(AA')c form a system of equiangular double spirals about A and 



