78 THEORY OF COLLINEATIONS. 
configuration in 2’; the range of points on PQ goes over 
into a projectively related range on P’@; also the range on 
PR goes over into a projectively related range on PR. 
Since the ranges on PQ and PR are projectively related, the 
corresponding ranges on P’Q and P’RF are also projectively 
related. These ranges are also in perspective position, since 
P, the self-corresponding point of the perspective ranges on 
PQ and PR, goes over into P’, which is therefore a self-cor- 
responding point of the ranges on P’Q and P’R. Since these 
ranges have a self-corresponding point, it follows that the 
lines joining their corresponding points meet ina point. This 
point is determined by the intersection of any two of the 
lines joining corresponding points. We know that in a the 
two tangents from P, to K cut PQ and PR in pairs of corre- 
sponding points. These tangents go over into the two tan- 
gents in 2’ fromP,’ to K’ and hence these tangents also join 
pairs of corresponding points on P’Q and P’R. Therefore, 
P/ is the vertex of the pencil in a’ projecting the range on 
P’Q into that on P’R. Let X and Y bea pair of correspond- 
ing points on PQ and PR respectively. The points P,, X, Y 
in z are collinear; their corresponding points in z’ are P,’, X’, 
Y’, which are also collinear, therefore our construction trans- 
forms straight lines in =z into straight lines in a’. 
Since our construction transforms collinear ranges on tan- 
gents to K into collinear ranges on tangents to K’, and every 
collinear range in x into a collinear range in 7’, it readily fol- 
lows that any range of points on a line g in z is transformed 
into a projectively related range on g’ in z’. Also any pencil 
of rays through a point P in z is transformed into a project- 
ively related pencil through P’ in a’. 
We have now proved that our construction by means of 
two conics K and K’ correlates the plane a to a’ in precisely 
the same manner as the perspective projection described in 
art. 87, except in the matter of self-corresponding points and 
lines; 7. €., it establishes a one-to-one correspondence be- 
tween the points of the two planes, between the lines of the 
