EMCH: INVOLUTORIC COIJ-INEATIONS IN PLANE AND IN SPACE. 21 7 



The parameter k' may, like k, assume any real value, since from 

 the expression above we deduce 



b^-aki. 



a — ck^' 



Considering the transformations (12) and (13) separately it is 

 seen that the planes through the z-axis are transformed into two 

 certain systems of planes through the z-axis by the transformations 

 (12), and (13), such that the one system is the reflection of the 

 of/uT oil the yz plane. 



The double-elements of (13) are 



:\/4 + k^^' 



c 



ki "" 

 ^ "b 



Designating them by r and s, there is 



b 



b 

 c 



P 



r+P 



s+P ~~ 

 s— p 



i. e., Tlie ill-ear iaiit planes of the resulting involutions {i^) form a 

 harnionie ratio 7oith the invariant planes of the transformation (7). 



If a^^; bc=i, the involutions in the planes z=±i become axial 

 symmetry and assume the form 

 x^=ax + by, 

 yi=cx— ay, 



and ^1= — a^ — ^^y? 



yi= — ex — ay. 



Every straight line intersecting the planes z= + i and z= — i in 

 two points A and A^ is transformed into a line passing through the 

 correspoinding points of A and A^ For intermediate points the 



relation zi=-^ must be added. Thus the geometrical interpre- 

 z 



tation of these involutions is easilv obtained. 



