This section should be taken as an introduction to the Ursell development. A more rigorous treatment appears in section 3.8. Substituting for the 
in equations 3.6-3.9 with 
, we have a hierarchy of functions 
written in terms of the reduced probability distribution functions. For example, 
. The functions are known as the Ursell functions. The relationship between the and the 
is known as the Ursell development. Note that the subscript on the indicates the order of the reduced density function, while the argument(s) indicate the variable(s) not averaged over. Note also, that the kernel 
may in actuality be any function W having been normalized so that summing over its arguments yields one, so in what follows W has been substituted for . Writing the reduced W functions in terms of the Ursell functions turns out to have a simple form, (the sum over all partitions).
Here each argument of the W functions is indicated by the argument's subscript.