4. Two boys walk along two paths *u* and
*u’* each holding a string which they keep
stretched tightly between them. They both move
at constant but different rates of speed, letting
out the string or drawing it in as they walk.
How many times will the line of the string pass over
any given point in the plane of the paths?

5. Trace the lines of the string when the two boys move at the same rate of speed in the two paths but do not start at the same time from the point where the two paths intersect.

6. A ship is sailing on a straight course and keeps a gun trained on a point on the shore. Show that a line at right angles to the direction of the gun at its muzzle will pass through any point in the plane twice or not at all. (Consider the point-row at infinity cut out by a line through the point on the shore at right angles to the direction of the gun.)

7. Two lines *u* and *u’* revolve
about two points *U* and *U’* respectively
in the same plane. They go in the same direction
and at the same rate of speed, but one has an angle
a the start of the other. Show that they generate
a point-row of the second order.

8. Discuss the question given in the last problem when the two lines revolve in opposite directions. Can you recognize the locus?

CHAPTER IV — POINT-ROWS OF THE SECOND ORDER

*60. Point-row of the second order defined.*
We have seen that two fundamental forms in one-to-one
correspondence may sometimes generate a form of higher
order. Thus, two point-rows (§ 55) generate a
system of rays of the second order, and two pencils
of rays (§ 57), a system of points of the second order.
As a system of points is more familiar to most students
of geometry than a system of lines, we study first
the point-row of the second order.

*61. Tangent line.* We have shown in the
last chapter (§ 55) that the locus of intersection
of corresponding rays of two projective pencils is
a point-row of the second order; that is, it has at
most two points in common with any line in the plane.
It is clear, first of all, that the centers of the
pencils are points of the locus; for to the line *SS’*,
considered as a ray of *S*, must correspond some
ray of *S’* which meets it in *S’*.
*S’*, and by the same argument *S*,
is then a point where corresponding rays meet.
Any ray through *S* will meet it in one point
besides *S*, namely, the point *P* where
it meets its corresponding ray. Now, by choosing
the ray through *S* sufficiently close to the
ray *SS’*, the point *P* may be made
to approach arbitrarily close to *S’*, and
the ray *S’P* may be made to differ in
position from the tangent line at *S’*
by as little as we please. We have, then, the
important theorem

*The ray at *_S’__ which corresponds to
the common ray __SS’__ is tangent to the locus
at __S’__._