  We generalise a formula of Shou-Wu Zhang, which describes local arithmetic
intersection numbers of three Cartier divisors with support in the special
fibre on a a self-product of a semi-stable arithmetic surface using elementary
analysis. By an approximation argument, Zhang extends his formula to a formula
for local arithmetic intersection numbers of three adelic metrized line bundles
on the self-product of a curve with trivial underlying line bundle. Using the
results on intersection theory from arXiv:1404.1623 [math.AG] we generalize
these results to d-fold self-products for arbitrary d. For the approximations
to converge, we have to assume that d satisfies the vanishing condition 4.7
from arXiv:1404.1623 [math.AG], which is true at least for $d\in \{2,3,4,5\}$.
