On spherical classes in $H_{\ast }QX$

We give an upper bound on the set of spherical classes in $H_{\ast }QX$ when $X = P,S^1$\mathrm{.}ThisisrelatedtotheCurtisconjectureonsphericalclassesin$H_*Q_0S^0$. The results also provide some control over the bordism classes on of immersions when $X$ is a Thom complex.