Statistical mechanical averages of vectors and tensors characterizing the configurations of polypeptides have been calculated for poly-L-alanines (PLA) of xu = 2-400 peptide units. These quantities are expressed in the reference frame of the first peptide unit, the X axis being situated along the virtual bond, the Y axis in the plane of the peptide unit. The persistence vector a identical to (r) converges rapidly with chain length to its limit a infinity which lies virtually in the XZ plane. Configurational averages of Cartesian tensors up to the sixth rank formed from the displacement vector p = r-a have been computed. For xu greater than 50 the even moments of fourth and sixth rank formed from the reduced vector p for the real chain are well repreented by the freely jointed chain with 21.7 virtual bonds equivalent to one of the model. The moments of p display assymmetry for xu less than 50. Density distribution functions Wa(p), evaluated from the three-dimensional Hermite series truncated at the term in the polynomial involving the tensors of p of sixth rank, display no obvious symmetry for xu less than 50. Approximate spherical symmetry of the distribution of p about a is observed only for xu greater than or equal to 100.