By Lester R Ford
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Extra info for Automorphic Functions
A transformation carries its isometric circle into the isometric circle of the inl'erse transformation. The inverse transformation, z' = (-dz b)/(cz - a), has the isometric circle I': Icz - al = 1. (45) + Its center is ale, its radius l/lcl. Now T carries I into a circle 1 0 without alteration of lengths in the neighborhood of any point) 26 LINEAR TRANSFORMATIONS [SEC. 11 hence T-l carries 1 0 back to I without alteration. But I' is the cOlnplete locus of points in the neighborhood of which T effects no change qf length; hence, 1 0 coincides with I'.
Let a point P on I be carried into P'. Then, if I be set down upon I' so that P coincIdes with P', with proper orientation, corresponding points will coincide. Any sequence of an even number of inversions which will effect the proper transformation on I will be equivalent to T (Theorem 7). FIG. 9. As a point moves from P covnterclockwise around I, suppose that the corresponding point moves from P' counterclockwise around I'. Then, I can be carried into I' by a rigid motion so that corresponding points coincide.
Vfe have, b = c, a = d, d = ii. These values obviously satisfy the conditions (a) and (b). -The most general linear transformation carrying Qo into itself and carrying the interior of Qo into itself is the transformation (47) aii-cc= 1. Z ' _- az + c cz + ii' The most general linear transformation carrying Qo into itself and carrying the interior of Qo into its exterior is found similarly. Then, -die is within Qo and dd - cc < 0; so that X = -1. The most general transformation is the resulting loxodromic transformation az - c z' - cz - ii' cc - aii = 1.
Automorphic Functions by Lester R Ford