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."

*10. Infinity-to-one correspondence.* It might be inferred that any
infinite assemblage could be put into one-to-one correspondence with any
other. Such is not the case, however, if the correspondence is to be
continuous, between the points on a line and the points on a plane.
Consider two lines which lie in different planes, and take _m_ points on
one and _n_ points on the other. The number of lines joining the _m_
points of one to the _n_ points jof the other is clearly _mn_. If we
symbolize the totality of points on a line by [infinity], then a
reasonable symbol for the totality of lines drawn to cut two lines would
be [infinity]2. Clearly, for every point on one line there are [infinity]
lines cutting across the other, so that the correspondence might be called
[infinity]-to-one. Thus the assemblage of lines cutting across two lines
is of higher order than the assemblage of points on a line; and as we have
called the point-row an assemblage of the first order, the system of lines
cutting across two lines ought to be