GEOMETRIC THEORY. 49 
59. Elliptic Transformations. Inthe case when the two 
tangents to K perpendicular to OX are imaginary, it still 
holds that the ecross-ratio of the four points of intersection of 
any tangent to K with the lines /, /’ and the two imaginary 
tangents AA’ and BB’ is constant. The two invariant points 
in this case are conjugate imaginary. A transformation with 
two conjugate imaginary invariant points is called an elliptic 
transformation. In this case the constant cross-ratio k is a 
complex number of the form e*’, 2. e., its modulus is unity. 
See exercise 8, page 61. This constant k in both hyperbolic 
and elliptic cases is called the characteristic cross-ratio of the 
transformation T. 
60. Parabolic Transformations.* But when the invariant 
points of the transformation coincide, we no longer have a 
characteristic cross-ratio for the transformation. However, 
another relation is found to hold for pairs of corresponding 
points, which relation is constant for all pairs of correspond- 
ing points in the transformation. We shall now proceed to 
determine this relation. It may be obtained in a very simple 
manner by considering the parabolic transformation as the 
limiting case of a hyperbolic transformation. 
We have (ABPP,) =k; hence (APBP,) =1—k. Writ- 
ing this out in full we get 
° —_s . = 1 a 
Pi PPE Z RIP, AE ANB 
When A and B coincide in (ABPP,) =k, we have k= 1; let 
eae 1-k ; 
the limit of 5 =; then 
BRR ee AR AP: a ee ip = fe t 
AP.AP, AP.APR  £AP, Let ye ie es 
1 1 
pian ape ine. (38) 
*The terms Hyperbolic, Elliptic, and Parabolic Transformations are due to Klein, 
and were first used by him in a paper entitled ‘* Ueber die Transformation der ellip- 
tischen Functionen und die Auflésung der Gleichungen 5* Grades,’’ Math. Annalen, 
Band 14, 1878. The names were suggested by the relations of the conic sections to 
the line at infinity. A hyperbola cuts the line at infinity in two real points, an 
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