GEOMETRIC THEORY. 51 
in the real quadrilateral ABB’A’ determines a system of 
transformations each of which is hyperbolic and leaves in- 
variant the two points A and B. These ~! transformations 
evidently constitute the hyperbolic group hG,(AB). The 
conics of the range touching the line / between A and B are 
ellipses; those touching the line / external to the segment 
AB are hyperbolas. 
If the two sides of the quadrilateral AA’ and BB’ are con- 
jugate imaginary lines, the inscribed range of conics deter- 
mines an elliptic group eG,(AB) whose invariant points are 
the conjugate imaginary points A and B. All conics of this 
range are hyperbolas. 
If the range of conics consists of hyperbolas having one 
common asymptote perpendicular to the bisector OX, (Fig. 4), 
the group is parabolic, pG,(A). 
THEOREM 31. The range of conics inscribed in a quadrilateral 
consisting of the lines? and// and a pair of lines perpendicular to 
the bisector of J and // determines a one-parameter group of trans- 
formations on the line 1. 
64. Resultant of Two Elliptic or Hyperbolic Transforma- 
tions. Let 7’, be the transformation of the group G,(AB), 
which transforms P to P,; thenk =(ABPP,). Let T,, be the 
transformation of the same group which transforms P, to P,; 
then k, = (ABP,P,). The two transformations T, and T,,, are 
together equivalent to a single transformation T;,, of the 
same group which transforms P to P,. To prove this we 
have 
AV AZ au, AA AU 
[ed ra i OCI, (Aub 2, 2) — me Saar 
Eliminating the fraction containing P, from those two equa- 
tions we have 
AP | AP, _ 
east) apo (ABE, ). (39) 
The conic of the range R whose tangential cross-ratio is kk,, 
gives a transformation which is equivalent to the combined 
