Now suppose that we have a point ato, at a distance s from the front surface of the glass, and another point O' at a distance s'inside the glass, and we desire to arrange the curved surface in such a manner that every ray from O which hits the surface, at any point P, will be bent so as to proceed toward the point o,. For that to be true, we have to shape the surface in such a way that the time it takes fo the light to go from O to P, that is, the distance oP divided by the speed of light (the speed here is unity), plus n. P, which is the time it takes to go from P to O is equal to a constant independent of the point P. This condition supplies us with an equation for determining the surface. The answer is that the surface is a very complicated fourth-degree curve, and the student may entertain himself by trying to calculate it by analytic geometry. It is simpler to try a special case that corre sponds to s-00, because then the curve is a second-degree curve and is more recognizable. It is interesting to compare this curve with the parabolic curve we found for a focusing mirror when the light is coming from infinit So the proper surface cannot easily be made--to focus the light from one point to another requires a rather complicated surface. It turns out in practice that we do not try to make such complicated surfaces ordinarily, but instead we make a compromise. Instead of trying to get all the rays to come to a focus, we arrange it so that only the rays fairly close to the axis 00 come to a focus. The farther ones may deviate if they want to, unfortunately, because the ideal surface is complicated, and we use instead a spherical surface with the right curvature at the axis. It is so much easier to fabricate a sphere than other surfaces that it is profitable for us to find out what happens to rays striking a spherical surface supposing that only the rays near the axis are going to be focused perfectly Those rays which are near the axis are sometimes called paraxial rays, and what we are analyzing are the conditions for the focusing of paraxial rays. We shall that troduced by the fact tha close to the axis he height P@ is h. For a moment, we imagine that the surface is a plane passing the through P. In that case, the time needed to go from o to P would exceed the time from 0 to 2, and also the time from P to o would exceed the time from g to o But that is why the glass must be curved, because the total excess time must be compensated by the delay in passing from V to g! Now the excess time along route OP is h/ 2s, and the excess time on the other route is nh2 /2s. This excess time, which must be matched by the delay in going along v@, differs from wha it would have been in a vacuum, because there is a medium present. In other yords, the time to go from V to e is not as if it were straight in the air, but it is slower by the factor n, so that the excess delay in this distance is then(n-1)vQ And now, how large is v@? If the point C is the center of the sphere and if its radius is R, we see by the same formula that the distance vg is equal to h2/2R. Therefore we discover that the law that connects the distances s and that gives us the radius of curvature R of the surface that we need t and s, (h27/2)+(mh2/2s)=(n-1)h2/2R (1/s)+(n/s)=(n-1)/R (273) If we have a position O and another position O', and want to focus light from O to o, then we can calculate the required radius of curvature R of the surface by this formula Now it turns out, interestingly, that the same lens, with the same curvature R, will focus for other distances, namely, for any pair of distances such that the sum of the two reciprocals, one multiplied by n, is a constant. Thus a given lens will(so long as we limit ourselves to paraxial rays)focus not only from O to out between an infinite number of other pairs of points, so long as those pairs of points bear the relationship that 1/s +n/s' is a constant, characteristic of the lens In particular, an interesting case is that in which s- oo. We can see from the formula that as one s increases, the other decreases. In other words, if point o