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.