134 KANSAS UNIVERSITY SCIENCE BULLETIN. 



Theorem 18. All transformations having a common invariant 

 point A, and for which the vahie of c in the formula k = exp(c + i)0 

 is the same, form a three-parameter subgroup H3(A)c of H^A) ; 

 eIL(A) and hHs(A) are among the subgroups H4(A). 



§ 7. Interpretation of He in Complex Plane. 

 The linear fractional transformation, 



T:zi = Sr (34) 



in which the coefficients and variables are complex quantities, is 

 capable of interpretation as a point transformation of the complex 

 plane. We proceed to the development of the transformation from 

 this point of view. 



T is a circular transformation. — We shall first show that the 

 transformation T transforms circles into circles. The equation of a 

 real circle in Cartesian coordinates is 



x? + yf + 2gxi + 2fyi + c = o. (35) 



Let xi -f iyi == z, xi — iyi = z", g + if = a, g — if = /?. Making these 

 substitutions in (34), we get as the equation of the circle : 



Zi z"i + /?Zi -f- ai"i -f y = o. (36) 



a and P are conjugate complex quantities; so, also, are zi and z^, but 

 y is real. 



If we change the sign of i throughout the equation of T, we have : 



a z + b 



Zi 



(37) 



c z + d 



where the quantities under the dash are the conjugates of those with- 

 out the dash. Substituting (34) and (37) in (36) and clearing, we 

 have : 



Kzz+Lz + Mz+N = o, (38) 



where K = aa -j- P a"c + a a~c + y cc , 



L = a b + P ad + a cb + y cd , 



M = ab + p cb + a ad + y cd, 



N == b b -f fi bd + a. b d + y dd. 

 On examination. K and N are found to be real and L and M con- 

 jugate imaginary. These are the necessary and sufficient conditions 

 that equation (38) represents a circle. Thus circles are transformed 

 into circles by T, which is, therefore, called a circular transformation. 



Straight lines are transformed into circles. — It may happen for 

 certain values of a, b, c, d that K in equation (38) vanishes. In this 

 case (38) represents a straight line. If we substitute (34) and (37) 

 in Ax -f- By + C==o, we find that this line is transformed into a circle. 



