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’__._ In the same manner the tangent at _S_ may be constructed.