perspectivity. A point-row and an axial pencil are in perspective position
if each plane of the pencil goes through the point on the point-row which
corresponds to it, and an axial pencil and a pencil of rays are in
perspective position if each ray lies in the plane which corresponds to
it; and, finally, two axial pencils are perspectively related if
corresponding planes meet in a plane.




*9. Projective relation.* It is easy to imagine a more general
correspondence between the points of two point-rows than the one just
described. If we take two perspective pencils, _A_ and _S_, then a
point-row _a_ perspective to _A_ will be in one-to-one correspondence with
a point-row _b_ perspective to _B_, but corresponding points will not, in
general, lie on lines which all pass through a point. Two such point-rows
are said to be _projectively related_, or simply projective to each other.
Similarly, two pencils of rays, or of planes, are projectively related to
each other if they are perspective to two perspective point-rows. This
idea will be generalized later on. It is important to note that between
the elements of two projective fundamental forms there is a one-to-one
correspondence, and also that this correspondence is in general
_continuous_; that is, by taking two elements of one form sufficiently
close to each other, the two corresponding elements in the other form may
be made to approach each other arbitrarily close. In the case of
point-rows this continuity is subject to exception in the neighborhood of
the point "at infinity."