and comes back, and all times must be equal. Here the light always travels in air so the time and the distance are proportional. therefore the statement that all the times are the same is the same as the statement that the total distance is the ame.Thus the sum of the two distances ri and r2 must be a constant. An ellipse is that curve which has the property that the sum of the distances from two points is a constant for every point on the ellipse; thus we can be sure that the light from one focus will come to the other The same principle works for gathering the light of a star. The great 200-inch Palomar telescope is built on the following principle. Imagine a star billions of miles away; we would like to cause all the light that comes in to come to 26-11 mirror Of course we cannot draw the rays that go all the way up to the star, but we still want to check whether the times are equal. Of course we know that when the vari- ous rays have arrived at some plane KK, perpendicular to the rays, all the times this plane are equal( Fig. 26-12). The rays must then come down to the mirror and proceed toward P'in equal times. That is, we must find a curve which has the k property that the sum of the distances Xr+ Xp"is a constant,no matter where X is chosen. An easy way to find it is to extend the length of the line XX down to a plane LL,. Now if we arrange our curve so that A'A= AP,BB"=Bpr C'C= CP, and so on, we will have our curve, because then of course, AA+ A'Pr= AA+ A'A will be constant. Thus our curve is the locus of all points equidistant from a line and a point. Such a curve is called a parabola; the mirror is made in the shape of a parabola The above examples illustrate the principle upon which such optical devic -t' can be designed. The exact curves can be calculated using the principle that focus perfectly, the travel times must be exactly equal for all light rays, as wel. o Fig. 26-12. A paraboloidal mirror. being less than for any other nearby path We shall discuss these focusing optical devices further in the next chapter let us now discuss the further development of the theory. When a new theoretical principle is developed, such as the principle of least time, our first inclination might be to say, Well, that is very pretty; it is delightful; but the question is, does it help at all in understanding the physics? "Someone may say, "Yes, look at how many things we can now understand!"Another says, "Very well, but I can under stand mirrors, too. I need a curve such that every tangent plane makes equal angles with the two rays. I can figure out a lens, too, because every ray that comes to it is bent through an angle time and the statement that angles are equal on reflection, and that the sines of the angles roportional on refraction, are the same. So is it merely a philo sophical question, or one of beauty? There can be arguments on both sides However, the importance of a powerful principle is that it predicts new things It is easy to show that there are a number of new things predicted by Fermat's principle. First, suppose that there are three media, glass, water, and air, and we perform a refraction experiment and measure the index n for one medium against another. Let us call n12 the index of air(1)against water( 2); n13 the index of air index, which we shall call n23. But there is no a priori reason why there should be any connection between n12, n13, and n23. On the other hand, according to the idea of least time, there is a definite relationship. The index nia is the ratio of two things, the speed in air to the speed in water; n13 is the ratio of the speed in air to the speed in glass; n23 is the ratio of the speed in water to the speed in gla Therefore we cancel out the air, and get n23 In other words, we predict that the index for a new pair of materials can be ob- tained from the indexes of the individual materials, both against air or against vacuum. So if we measure the speed of light in all materials, and from this get a ingle number for each material, namely its index relative to vacuum, called n