Now let's look at another way we can use this tabular paraxial ray tracing. The two previous examples I did used numerical descriptions of the rays and traced those rays numerically through the system, so that we could answer numerical questions about the system. But it's so mechanical, we could also do the same thing symbolically. And this is now our first real powerful design tool, because you can now take and describe a ray, some important ray that you've chosen cleverly by it's height and its angle as symbols and push that ray through the system to come up with design equations. So let's do that for what's called a Keplerian telescope, invented by, of course, Johannes Kepler. It consists of two positive focal length lenses that in the simplest case are separated by the sum of their focal lengths. And this is such a common system that it's worth understanding and just having in your back pocket, because you will use it a lot as an optical designer. It's also one we could trace very easily by graphical ray tracing. But let's just do it with the y-u method to see what it tells us. And I have a bit of an abbreviated table down here because we're getting good at this at this point. I'm going to launch a ray from object space that's at a particular height h1 and is axial ray. So it's coming in with angle 0. And then at surface 1, the first lens,and surface 2, the second lens, I'm going to calculate how that ray evolved through the system but symbolically. So, for example, when I get behind surface 1 here, the ray height is still h1 because I just finished going to that surface. But the ray angle through my refraction equation will be minus h-1 times the power of lens 1 or equivalently divided by f1. And again, you can just see that by our little triangle from graphical ray tracing. It gets little more interesting now when we transfer from lens 1 to lens 2 by a distance d which is the sum of the focal lengths. So first, let's do the transfer equation. That's the top one here. And that says that our ray height changes by, we add a height that is proportional to the ray angle and multiplied by the distance we traveled. So that there is the ray angle h1/f and then multiplied by the distance we traveled f1 plus f2. And if you simplify that, you notice that the new ray height which I will call h2 out here, is the old ray height times minus f2/f1, which again we could have just gotten by these similar triangles. So we just learned that the magnification of this system is given by the ratio of the focal lengths and that's a nice thing to know. Now if we go ahead and refract through lens f2 by a refraction equation, we start with our incoming angle and then we change it by the ray height times lengths to power. So that's this expression here. And when we simplify that it turns out the ray height, I'm sorry, the ray angle comes out to be 0. And that's what we, again, we would expect from our sample ray tracing but we've now proven that mathematically. And of course, if you gave their self a distance here d that did not equal the sum of the focal lengths, this ray angle would not be 0 and that would like to start, for example, designing a zoom lens in which you want to understand how the ray height and ray angle coming out of a two length system depend on the length separation. And these equations that are what we can use to start writing down design equations for complex lens elements like the camera we saw at the beginning of the course, where we start moving lens bundles or lens packages around. So what we have proven from this system which we could have done graphically is that if we have two positive focal length lenses separated by the sum of their distances, it's a focal. That is a ray that comes in parallel to the axis, exits parallel to the axis. And we've proven that the magnification of the system is minus f2/f1 which is something you'll use so often it's probably worth remembering. And just for fun, just to emphasize why all this baggage and design convention and equations are useful, what if we wanted to evaluate a Galilean telescope? Galileo noted that we could make the system better in a couple of ways if we could grind a negative lens. Remember this symbol here with the arrows pointing inward is our symbol for a negative lens. In that case, we can shorten the system because we can still make this a focal system. And if you don't immediately see why this makes sense, stop and do a graphical ray trace. We can get the same afocal system. We can get the same magnification to a design but the system is smaller where it's shorter for the same magnitudes of the focal lengths. And the image is upright. And that's kind of neat. It doesn't turn everything upside down. So I could make a new table and I can trace all the way through this again or I could notice my sign convention does this for me. Because f2 can simply be replaced by a negative number and then all the same equations, the positions and angles of the rays, the object and image positions, even the height that this object becoming positive all work, because f2 is not negative. And that is a really good example of why the sign convention is our friend. d here is now f1 minus a number because f2 is a negative number and that just shows us that the system is more compact. So this is why the same convention is worth it. We've just gone from a Keplerian telescope to a Galilean and we realized they're absolutely the same thing, we just changed the sign of one of the focal lengths.