The quantal cross section Q(qm)((N)) for collision in dilute gases is given for any order N, where previously in the literature only values up to N=4 were to be found. The quantum-statistical character of Q(qm)((N)) is discussed and calculations. are presented for Q((1)) through Q((6)) for Boltzmann, Fermi-Dirac, and Bose-Einstein cases for He-3-He-4, He-3-He-3, and He-4-He-4 using extensive tables of phase shifts.