Elementary Functions 235Fig. 5.9
Oc. Figure 5.10 represents (assuming that z 1 = 0 for simplicity) the two
corresponding families of orthogonal circles into which the two families of
Fig. 5.9 are transformed when returning to the original variables z, w. Note
that the line OA, which contains the origin, goes into the symmetric line
0 A'. Similarly, 0 B ..L 0 A goes into 0 B' ..L 0 A'. The lines parallel to 0 A
in the upper half-plane are transformed into circles tangent to OA' at 0
but lying in the lower half-plane. The lines parallel to OB are carried into
the circles tangent at 0 and orthogonal to the circles of the first family.
In the preceding discussion (from which the identical transformation has
been excluded) we have seen that if the bilinear transformation is of one
A'Fig. 5.10