  The O(N) Heisenberg and spherical models with interaction given by the long
range hierarchical Laplacean are investigated. Two classical results are
adapted. The Kac--Thompson solution [KT] of the spherical model, which holds
for spacially homogeneous interaction, is firstly extended to hierarchical
model whose interaction fails to be translation invariant. Then, the
convergence proof of O(N) Heisenberg to the spherical model by Kunz and Zumbach
[KZ] is extended to the long range hierarchical interaction. We also examine
whether these results can be carried over as the size of the hierarchical block
L goes to 1. These extensions are considered a preliminary study prior the
investigation of the model by renormalization group given in [MCG] where
central limit theorems for the spherical (N=\infty) model on the local
potential approximation (L\downarrow 1) are then established from an explicit
solution of the associate nonlinear first order partial differential equation.
