94, 95.] TRANSFORMATION OF COMPLEX VARIABLES. 101 



To solve the above problem we remark in the first place that 

 any assumption of the form 



, AZ+B _ -DZ +B 

 Z= ~(JZVD&amp;gt; r Z = CZ -A ............ ( 3) 



transforms circles into circles. For suppose the point Z to de-. 

 scribe a circle about any point C as centre. We have then 



mod (Z r - (7) = const., 



and therefore, if by a change of constants Z G be put into the 

 form 



- 

 Z-C, 



mod (Z- 0,) : mod (Z- &amp;lt;7J = const. 



Hence the point Z moves so that its distances from the two 

 points Cj, C 2 are in a constant ratio; i.e., by a well-known theorem 

 of elementary geometry, it describes a circle. To a lune in the 

 plane of Z corresponds then a lune of the same angle in the 

 plane of Z . The three ratios A : B : C : D may be so chosen as 

 to make any three points in the one plane correspond to any 

 three points in the other, when the circles determined by these 

 triads will also correspond. Hence to establish a correspondence 

 between any two limes of the same angle we have only to de 

 termine the above ratios so that the angular points of the one 

 shall correspond to the angular points of the other, and any third 

 point on the perimeter of the one to any third point on that of 

 the other. 



As a particular case, the assumption 



transforms any lune Tvhose angular points are at c x , C 2 into a lune 

 in the plane of 5% having its angular points at and oo , i.e. into 

 two straight lines radiating from the origin, and making an angle 

 equal to that of the lune. 



If we now assume 



Z = 5S&quot;, 



these straight lines become, in the plane of Z, straight lines 



