This is what I am writing::: START at the point y=0, just at the ring’s edge, where we say r=k. Here, on the outside axis, r^2 = x^2 -1. . Since the E-fields are in Kerr coords, we differentiate wrt /x,y/ : dE = ∂E/∂r ∂r/dx + ∂E/∂theta ∂theta/dx… for both components, /x,y/ separately; also we process both field components, at least with increasing ‘y’. At the starting edge point, the sqrt gives r= sqrt(x^2 – 1) , and with k^2 = [x_0^2] – 1 , we can write ‘x’_0 as sqrt[ 1 + k^2 ] ≈ 1 + 1/2 k^2 . OK, this is the RING POINT, my TINKER BELL POINT. Now I expand the two field components, around THIS POINT.
I think Steve will see this process yielding my TINKER BELL EQUATIONS.
Moving a bit in the equatorial plane with a ‘dx’, clearly only the radial part changes. Moving with a ‘dy’ is more subtle.