  We investigate, using finite size numerical calculations, the spin-polarized
fractional quantum Hall effect (FQHE) in the first excited Landau level (LL).
We find evidence for the existence of an incompressible state at $\nu =
\frac{7}{3} = 2+\frac{1}{3}$, but not at $\nu = 2+\frac{2}{5}$. Surprisingly,
the 7/3 state is found to be strongest at a finite thickness. The structure of
the low- lying excited states is found to be markedly different from that in
the lowest LL. This study also rules out FQHE at a large number of
odd-denominator fractions in the lowest LL.
