  In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain
some new nonimmersions of real projective spaces. In this note, we
complete/correct two oversights in that paper.
  The first is to note that in that paper a general nonimmersion result was
stated which yielded new nonimmersions for RP^n with n as small as 48, and yet
it was stated there that the first new result occurred when n=1536. Here we
give a simple proof of those overlooked results.
  Secondly, we fill in a gap in the proof of the 2002 paper. There it was
claimed that an axial map f must satisfy f^*(X)=X_1+X_2. We realized recently
that this is not clear. However, here we show that it is true up multiplication
by a unit in the appropriate ring, and so we retrieve all the nonimmersion
results claimed in the original paper.
  Finally, we present a complete determination of tmf^{8*}(RP^\infty\times
RP^\infty) and tmf^*(CP^\infty\times CP^\infty) in positive dimensions.
