Legendre's formula for the surface area of scalene ellipsoid

Question

I am looking for a derivation of the formula $$S=2\pi c^2 +\frac{2\pi a b}{\sin \varphi}\left(E(\varphi, k)\sin ^2 \varphi +F(\varphi,k)\cos^2 \varphi\right)$$ for the surface area of the ellipsoid $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ with $$a>b>c, \hspace{3ex} \cos \varphi=\frac{c}{a}, \hspace{3ex} k^2=\frac{a^2(b^2-c^2)}{b^2(a^2-c^2)},$$ where $F(\varphi,k)$ is Legendre's incomplete elliptic integral of the first kind and $E(\varphi,k)$ is Legendre's incomplete elliptic integral of the second kind. This formula is simpler (and more elegant) than what one gets by integrating $$\int_0^a \int_0^{b\sqrt{1-x^2/a^2}}\sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2}dydx.$$ The formula was derived in a long and complicated way in an 1825 paper by Legendre, and it is often quoted (see for example the Wikipedia entry for ellipsoid), but I cannot find a proof.