Series Law for Extremal Length/Rho is Well Defined

To see that $\rho$ is a well-defined metric, we need to check that it transforms correctly when changing local coordinates.

Let $z=z(t)$ and $w=w(t)$ be charts on the Riemann surface $X$.

Let $\rho_1^z(t)$ and $\rho_1^w(t)$ be the coefficient functions when $\rho_1$ is expressed in the local coordinates $z$ and $w$, respectively.

We use the analogous notation for $\rho_2$ and $\rho$.

Since $\rho_j$ is a metric ($j\in\{1,2\}$), we have

$ \rho_j^w(t) = \rho_j^z(t) \cdot \left| \dfrac{dz}{dw}\right|$

(Here $\dfrac{dz}{dw}$ denotes, as usual, the derivative of the coordinate change $z\circ w^{-1}$.)

Thus we have:

$\displaystyle \rho^w(t) = \max(\rho_1^w(t),\rho_2^w(t)) = \max(\rho_1^z(t),\rho_2^z(t))\cdot \left| \frac{dz}{dw}\right| = \rho^z(t)\cdot \left| \frac{dz}{dw}\right|$

This means that $\rho$ transforms correctly and is a metric, as desired.