74 THEORY OF COLLINEATIONS. 
ence established by perspective projection is made perfectly 
general; compare Chapter I, article 46. 
89. Perspective Ranges and Pencils. Let p be a plane 
through P cutting / in O, and xand a’ inaand a’ respectively. 
The ranges of points on a and a’ are in perspective position 
and projectively related as defined in article 51; their point 
of intersection O is a self-corresponding point of the two 
ranges. 
The pencil of planes intersecting in the line PO is cut by 
and x’ in two pencils of lines which have a one-to-one corre- 
spondence, corresponding lines of the two pencils being 
coplanar with PO. Their common line / is a self-correspond- 
ing line of the two pencils. Two pencils of rays which are 
the sections of a pencil of planes by two other planes are said 
to be in perspective position and are projectively related. 
90. Self-corresponding Points and Lines. The aggre- 
gate of all points in a plane is called a field of points and the 
ageregate of all lines is called a field of lines. When two 
fields of points are connected by a perspective projection all 
points common to the two fields, 7. e., all points on the line J, 
are self-corresponding points of the two fields. When two 
fields of lines are connected by a perspective projection the 
common line | is a self-corresponding line of the two fields. 
The line / and all points on it are the only self-corresponding 
elements of the two fields. 
91. Invariance of Cross-ratios. The cross-ratios of any 
four collinear points in a and of their four corresponding 
points in z’ are equal. This follows from the fact that if four 
points in a lie on a line, say a, their four corresponding 
points in z’ lie on a’ which meets a in O, a point on/. The 
ranges on a and a’ are projectively related and in perspective 
position, and the invariance of cross-ratios of corresponding 
points was proved in art. 48, Chap. I. 
In like manner the cross-ratio of any four concurrent lines 
in x is equal to that of their four corresponding lines in 7’, 
