NORMAL FORMS. 117 
Rig. 22: 
of these invariant pencils there is a one-dimensional parabolic 
transformation having / for its single invariant line. 
Let S’ be an elation leaving invariant the fundamental 
figure of Fig. 22. A line g parallel to / will be transformed 
into g, also parallel to/; for g and g, both belong to a pencil 
whose vertex is the point at affinity on /. The line at infinity 
will be transformed by S’ into some lineas g;. Let us first 
consider here the parabolic transformation along the line OP 
perpendicular to .; we have, Art. 60, 
erie 1 1 1 eee 
ODO) an OP: = OP; 
The characteristic constant ¢ is the reciprocal of the seg- 
ment OP, where P, is the point into which the point at infinity 
