  This paper contains the proofs of Theorems 2 and 3 of the article entitled
Random Electrical Networks on Complete Graphs, written by the same authors and
published in the Journal of the London Mathematical Society, vol. 30 (1984),
pp. 171-192. The current paper was written in 1983 but was not published in a
journal, although its existence was announced in the LMS paper. This TeX
version was created on 9 July 2001. It incorporates minor improvements to
formatting and punctuation, but no change has been made to the mathematics.
  We study the effective electrical resistance of the complete graph $K_{n+2}$
when each edge is allocated a random resistance. These resistances are assumed
independent with distribution $P(R=\infty)=1-n^{-1}\gamma(n)$, $P(R\le x) =
n^{-1}\gamma(n)F(x)$ for $0\le x < \infty$, where $F$ is a fixed distribution
function and $\gamma(n)\to\gamma\ge 0$ as $n\to\infty$. The asymptotic
effective resistance between two chosen vertices is identified in the two cases
$\gamma\le 1$ and $\gamma>1$, and the case $\gamma=\infty$ is considered. The
analysis proceeds via detailed estimates based on the theory of branching
processes.
