NKWSON : ONK-DIMENSIONAL TRANSFORMATIONS. 135 



Hence, ciroles are sometimes transformed into a straight Line, and 

 straighl lines usually into circles. 



In the geometry of the complex plane, a straight line is regarded 

 as a special case of a circle, and every circle passing through the one 

 point at infinity of the complex plane is a straight line. Regarding 

 a straight line as a particular case of a circle, our transformation 

 always transforms circles into circles. 



Theorem 19. A linear fractional transformation in complex varia- 

 bles is a circular transformation of the complex plane. 



T is a conformal transformation. — Another fundamental property 

 of our transformation T is that angles are transformed into equal 

 angles. To show this, consider a small triangle formed by the points 

 z, z', z", and compare it with the triangle formed by their correspond- 

 ing points zi, z'i, x"i. In z z ',' ~ z z ' substitute their values from (34) ; 

 we get : 



z', — z, cz" + d z' 



(39) 



z"j — z t cz' + d z" — z 



The ratio c *', + ^ tends to unity as the triangles become very small. 

 The quantities z'i — zi, z' — z, etc., are the corresponding sides of the 

 two triangles; hence, when the corresponding triangles are very 

 small, the corresponding sides are proportional and the triangles are 

 similar. Angles are therefore transformed into equal angles. A 

 transformation which leaves angular magnitudes unaltered is called a 

 conformal transformation. 



Theorem 20. A linear fractional transformation of the complex 

 plane is a conformal transformation. 



T is a direct circular transformation. — The two triangles, z, z', z" 

 and zi, z'i, z"i, are not only in the limit similar triangles, but they are 

 also congruent triangles. The homologous sides are arranged in the 

 same order around the two triangles, and if they were of the same size 

 the two triangles could be made to coincide. 



Suppose that the three sides of the triangle z, z', z" are colored red, 

 green, and blue, and that an observer at z, facing the opposite side, 

 has on his right the red, on his left the blue, and before him the 

 green. The transformation T changes the triangle z, z', z" into the 

 new triangle zi, z'i, z"i, and the observer at his new position at zi still 

 finds the red on his right and the blue on his left. Such a trans- 

 formation is a direct circular transformation. 



There are other transformations which are both circular and con- 

 formal, but which interchange the parts of the triangle, so that in his 

 new position the observer finds the blue on his right and the red c.u 

 his left. These are called indirect circular transformations. 



