High-temperature exchange third virial coefficient for hard spheres via an asymptotic method for path integrals

The exchange part of the third cluster integral can be divided into two parts:b ₃(exch-1), which arises from the exchange of two particles, andb ₃(exch-2), which arises from the cyclic exchange of all three particles. The first few terms ofb ₃(exch-1) are calculated by arguing thatb ₃(exch-1) =-9a³/(4³)]b₂(exch)1 + O(/a)], whereb ₂(exch) is the exchange second cluster integral, is the thermal de Broglie wavelength, anda is the hardsphere diameter. The first three terms ofb ₃(exch-2) are calculated by writing it in path integral form and expanding about the shortest path.