Ursell Functions of the WN

It is readily seen that if any group of the locations in (1. N) is widely separated from all the rest, then decomposes into a product of the Boltzmann factors for the separated parts, and goes to zero. We say that the 

Boltzmann factors have the product property, and the Ursell functions the cluster property. 

It will be recalled that in multi(x mponent systems the bonds in graphs such as those in Fig. 6 are tensors. For a two-component system, for example, the graph consisting of a W3 bond joining three black z circles is given in terms of 

Returning to Eq. (80) we notice that another way to specify the Ursell functions is with the boundary condition W i(l) = %i(l) and the recursion formula 

The second equation is the same as Leibnitz s rule for the differential operator. The coordinate arguments of the Ursell functions have been suppressed in Eq. (86). 

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