  We determine the full structure of the leading (double-pole) divergences of
O(p^6) in the meson sector of chiral perturbation theory. The field theoretic
basis for this calculation is described. We then use an extension of this
result to determine the p^6 contributions containing double chiral logarithms
(L^2), single logarithms times p^4 constants (L x L_i^r) and products of two
p^4 constants (L_i^r x L_j^r) for F_\pi, F_K/F_\pi, K_{l3} and K_{e4} form
factors. Numerical results are presented for these quantities.
