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.