GEOMETRIC CONSTRUCTION. 81 
K’, the configuration in a’ corresponding to any given config- 
uration in a can be constructed. And further, one of the 
planes as a’ may be revolved about lJ until it coincides with a, 
and then the constructions all thought of as in the same 
plane. We shall make constant use of this last conception. 
By means of a non-perspective projection a field of points 
in x’ can be constructed corresponding to a given field in 2%. 
By a revolution about / this new field of points may be 
brought back to a and both fields of points thought of as ex- 
isting in the same plane. This operation of projecting the 
points in x into a new system of points in 7’ and, by revolv- 
ing about lJ, bringing the new system back to 2 will be called 
a projective transformation or collineation of the plane 2. 
Such a projective transformation is determined and com- 
pletely constructed by means of two conics K and K’ in the 
plane ~ and touching a line 1. 
THEOREM 11. A projective transformation or collineation of 
the points and lines of a plane is completely determined by means of 
two conics K and K’ both touching a fixed line / of the plane. 
N 
*The lines joining corresponding points of the planes z and x’ form a linear con- 
gruence (Strahlen-Congruenz) of the third order and first class. See Reye’s Geo- 
metrie der Lage, II. Band, p. 94. 
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