  We present a detailed evaluation of $\eta$, the critical exponent
corresponding to the electron anomalous dimension, at $O(1/N^2_{\!f})$ in a
large flavour expansion of QED in arbitrary dimensions in the Landau gauge. The
method involves solving the skeleton Dyson equations with dressed propagators
in the critical region of the theory. Various techniques to compute massless
two loop Feynman diagrams, which are of independent interest, are also given.
