through _S_. It is called _a pencil of planes of the second order_.




PROBLEMS


*1. * A man _A_ moves along a straight road _u_, and another man _B_ moves
along the same road and walks so as always to keep sight of _A_ in a small
mirror _M_ at the side of the road. How many times will they come
together, _A_ moving always in the same direction along the road?

2. How many times would the two men in the first problem see each other in
two mirrors _M_ and _N_ as they walk along the road as before? (The planes
of the two mirrors are not necessarily parallel to _u_.)

3. As A moves along _u_, trace the path of B so that the two men may
always see each other in the two mirrors.

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.