  For any analytic self-map $\psi$ of $\{z: |z| < 1\}$, J. H. Shapiro has
established that the square of the essential norm of the composition operator
$C_\psi$ on the Hardy Space $H^2$ is precisely $\limsup_{|w|\rightarrow
1^-}N_\psi(w)/(1-|w|)$; where $N_\psi$ is the Nevanlinna counting function for
$\psi$. In this paper we show that this quantity is equal to
$\limsup_{|a|\rightarrow 1^-}(1 - |a|^2)||1/(1 - \bar{a}\psi)||_{H^2}^2.$ This
alternative expression provides a link between the one given by Shapiro and
earlier measure-theoretic notions. Applications are given.
