6o 



KANSAS UNIVERSITY QUARTERLY. 



l_iJLlL 



Let S' be an elation leaving 

 invariant the fundamental figure 

 of Fig. 9. A line g parallel to 1 

 will be transformed into g, also 

 parallel to 1. For g and g^ both 

 belonging to a pencil whose 

 vertex is the point at infinity on 

 1. Some line as / will be trans- 

 formed by S' into the line at 

 infinity. Let us first consider 

 the parabolic transformation 

 along the line OP perpendicular 

 to 1; we have 



Fig. 9. 



I 

 OR 



I 

 00 



OP, OP OPi 



OP OPj 00 OPi ' ■ 



The characteristic constant a is the recipro,cal of the segment 

 OP; where Pj is the point that is transformed to infinity. Along 

 any other line through O as OS making an angle 6 with 1 we 

 have 



os;"4=^olr---"^- 



Let A any point on 1 be the vertex of an invariant pencil of rays 

 and let AO^^d. The elation S' transforms AP into AP, and AP, 

 into AK perpendicular to 1. Let the angle PAO=:<^, P^AO^^<j>^, 

 PjAO=<^i, etc. Along OP we have 



I I I 



" OP ~ ' 



I cot^i I COt<^ I 



OP , ^ d ' OP "" "d '. O P"i' 

 we have 



COt<^, — COt^;— da. (^4) 



Thus we have the expression for a one-dimensional parabolic 

 transformation of the pencil of lines through a point on 1. 



Theoreui j. Every parabolic transformation in the plane ivliether 

 along an invariant line or through an invariant point can be expressed 

 in terms of a single characteristic constant a. An elation is completely 

 determined bv its fundamental invariant figure and a characteristic 

 constant a. 



26. Type V a Special Case of Type II. We showed in the 

 last section how type IV might be considered as a special case of 



But 



Substituting these values 



