 Now, here comes your second curvature equation. Remember the equation, therefore, acceleration that we have from there. I'm going to do a silly thing, which will look silly. I'm going to take the cross product of those two vectors. I'm not writing, it's a function of t, it's a function of t. Just saving some space. I'm going to take that cross product. What's the cross product of this sum of these two components? Well, that's going to be the cross product of the first one, dT squared. t prime of t plus, I'm going to have here t cross product dT squared in of t. Oh, so I've put the t there. Now, what is the cross product? This is just the scalar. What is the cross product of two of the exact same vectors there? Well, that's zero, even though there's a scalar in front of them. The cross product of those two is going to be zero. So, we dropped this bit. And what is t cross n? t cross n, well, we know that's b. We know that's b, but that's a unit vector. Now, across a unit vector, that's going to be another unit vector. And its magnitude, if I suddenly were to say, well, let's not do that. Let's do this. Let's get the magnitude of this. So, I'm going to be left with this cross set, which is b, but I'm taking its magnitude, which is just one. What I'm left with, basically, is dS dT squared. And what else did I forget? Oh, I forgot the capital there. That looked a bit odd. And I'm left with the capital there. Now, we just have to remember a few things. What is t? Shall we do that one? What can we change? What can we change? Let's play with dS dT. dS dT, well, dS dT like that. That is just going to be the norm of r prime of t like that. So, we can put that in as square there. Let's put that in. We have this r prime of t squared. We've got that bit. And let's do something about this side. What can I write that as? Remember, I'm dealing here with the magnitude. So, remember that that was the r prime of t divided by its magnitude. That was that one. Cross, but I'm dealing all the time now with magnitudes. That was that, which was just the r prime of t. That's a. So, I'm crossing these. And that's going to equal kappa. And that's going to equal kappa norm of r prime of t there. But here, I'm dealing with scalars because I'm taking the magnitude. So, it would be the same as doing this. So, and behold, out jumps my second equation for kappa, which is going to be the velocity vector there. Cross product with the second derivative of the position vector there. Divided by the first derivative there of the position vector cubed. It's getting kappa. Sorry about that. It's getting kappa on its own. So, that's a very simple equation by someone who just played a bit. I suspect there was some deeper thought than that though. Going on, just taking that cross product there and just replacing these with values that we know what they are. And we're just taking magnitudes so we can view these more algebraically. And if I have a position vector, all I need now is a position vector. If I take its first derivative and I take its second derivative, that's all I need. That's all I need. If I do a cross product between those two vectors and I divide it by and I take its magnitude and I divide it by the magnitude of the first derivative cubed, I have curvature again. So, there's my second equation for curvature and it's actually quite beautiful.