80 THEORY OF COLLINEATIONS. 
projection. We wish to show that the projection is thereby 
uniquely and completely determined. 
Connect AB, BC, CD, and DA, by lines which we shall call 
a, b, c,d, respectively. The lines joining corresponding points 
are their corresponding lines a’, b’,c’,d’, respectively. The 
lines a,b,c,d,l in a and a’,b’,c’,d’,l/ in a’ determine the 
- conics K and K’ respectively; and these conics determine a 
non-perspective projection of x on x’. 
But the four points ABCD determine six lines, and these 
taken four at a time give us fifteen quadrilaterals. These 
fifteen quadrilaterals give rise to fifteen different pairs of 
conics, which determine either fifteen different projections of 
aon x’, or the same projection in fifteen different ways, or 
more than one and less than fifteen, some of them being du- 
plicated. 
Let us consider the lines AB, AC, etc., in a and their cor- 
responding lines A’ B’, A’C’, etc., ina’. The intersections of 
opposite sides of the quadrangle A’ B’C’ D’ correspond to the 
intersections of the corresponding opposite sides of the quad- 
rangle ABCD. In this way three new pairs of corresponding 
points are determined. New quadrangles may be formed 
out of these seven points in each plane, and thus other pairs 
of corresponding points obtained; and so on indefinitely. 
Hence when four pairs of corresponding points are given in 
the two planes, an unlimited number of pairs of correspond- 
ing points are determined. These considerations show that 
four pairs of corresponding points in the two planes determine 
one and only one non-perspective projection of a on a’. It 
will be shown later that there are * different constructions 
of the same non-perspective projection. 
THEOREM 10. Four pairs of corresponding points in the most 
general position are necessary and sufficient to determine a non-per- 
spective projection of one field of points on another. 
97. Projective Transformation. Since the constructions in 
the two planes are exactly alike, the operations of the last 
article are strictly reversible. By means of the conics K and 
