(This all started when I started to try to figure out how this first link below represents the ellipsoid with legendre polynomials)
On this web page regarding potentials outside of a sphere, it begins by stating that we will represent the ellipse we are to rotate around the z-axis, with mean radius ($a=\frac{r_{minor}+r_{major}}2$ I think, although we could use the geometric mean?) as
$$r=a_{\theta}(\theta)=a(\ 1-\frac{2}{3}\epsilon P_2(\cos{(\theta)})\ )$$ where $P_2(x)$ is the second legendre polynomial, $P_2(x)=\frac{3}{2}x^2-\frac{1}{2}\implies P_2(\cos{(\theta)}=\frac{3\cos^2{(\theta)}}{2}-\frac{1}{2} $ Substituting, we get $$r=a_{\theta}(\theta)=a(1-\frac{2 \epsilon}{3}(\frac{3}{2}\cos^2{(\theta)}-\frac{1}{2})=a-\frac{2}{3}\frac{3}{2}a \epsilon \cos^2{(\theta)}+\frac{a\epsilon}{3}=\frac{a\epsilon}3+a-a \epsilon \cos^2{(\theta)}=\frac{a\epsilon}{3}+a(\frac\epsilon3+1-\epsilon\cos^2{(\theta)})$$.
However, on wikipedia,with $\alpha$=semi-major axis, $\beta$= semi- minor axis,and eccentricity uses the same symbol as above, we are given the formula $$r(\theta)=\frac{\alpha\beta}{\sqrt{1-\epsilon^2 \cos^2{(\theta)}}}$$, expanding $\epsilon$, we get $$r(\theta)=\frac{\alpha\beta}{\sqrt{1-(\sqrt{1-\frac{\beta^2}{\alpha^2}})^2\cos^2{(\theta)}}}=\frac{\alpha\beta}{\sqrt{1-1+\frac{\beta^2}{\alpha^2}\cos^2{(\theta)}}}=\frac{\alpha\beta}{\sqrt{1+\frac{\beta^2}{\alpha^2}\cos^2{(\theta)}}}$$
I'm not really seeing how to connect these two, nor am I sure which definition of $a$ is used in the first set of equations. Any suggestions? I'd rather not have it just given to me, But i'm not really sure in which direciton I should go, including what form of mean was intended. It's this "mean radius" that I've never heard of and couldn't find a standard definition with searching google. I appreciate the help in advance :)