Stanley and Kaplan' have recently described a method for calculating high-temperature expansions for the specific heat, C, and susceptibility, y, of the Heisenberg ferromagnetic model in the limit of infinite spin. We had ourselves been working on this model, and had in fact obtained eight terms in the susceptibility series for the three cubic lattices. We wish to present these results together with tentative conclusions from a preliminary analysis of them.The method we used is not quite the same as that of Stanley and Kaplan, though it is fairly closely related to it. If we start with the graphical expansion of Rushbrooke and Wood' (i.e. , involving linear graphs, connected or disconnected, and with the possibility of more than one interaction line between each pair of vertices) and take the partial trace (of the sum of all terms corresponding to this graph in the expansion of the partition function) over the spin variables at all vertices of the graph except one, then, since the trace factor at any vertex is a scalar and the Heisenberg Hamiltonian is itself a scalar (with respect to rotation of spin axes) this partial trace must also be a scalar. For large spin we can ignore the commutation relations, and therefore this partial trace can be written Qa S S S pqr1 2 3' where (S"S"SS) is the spin vector at the remaining vertex (over which we have not yet taken the trace) Here p. +q+r equals I, the number of lines entering this vertex. Since this is a scalar it must be of the form f(X) where X=S(S+I) and S is the spin value. We now observe that to obtain the highest power of X, which is all we are concerned within the classical limit, l, indeed p, q, and r, must be even, and therefore the above partial trace is effectively a(S,S, + S,S, + S,S, )f/2, i.e. , aXE/2. Therefore, to find a we need consider only the term with l factors Sy at the chosen vertex.This has two consequences. First, it shows that the trace of any graph is simply the product of the traces of all its component parts, where to reduce a graph to its component parts we simply cut it at all its articulation points. Secondly, it provides a very easy method of computing the trace of any such component part (what Stanley and Kaplan call a classical diagram), for the spins associated with each end of all interaction lines entering any conveniently chosen vertex can all be replaced by S,. The trace calculations are in fact quite trivial, and do not involve integrations.We have presented the above argument in terms of graphs without crosses because the trace of a susceptibility graph with two crosses is simply related to that of the graph in which these are replaced by an extra interaction line (see Theorem IV of Rushbrooke and Wood' ). This is equivalent to Stanley and Kaplan's use of the spin correlation function.Having enumerated the graphs and found the requisite traces, we have still to determine the coefficient of N (the number of lattice sites) in the corresponding occurrence factors (for susceptibility graphs, not necessarily connected) on a physica...