  In the present paper, fixed trace $\beta$-Hermite ensembles generalizing the
fixed trace Gaussian Hermite ensemble are considered. For all $\beta$, we prove
the Wigner semicircle law for these ensembles by using two different methods:
one is the moment equivalence method with the help of the matrix model for
general $\beta$, the other is to use asymptotic analysis tools. At the edge of
the density, we prove that the edge scaling limit for $\beta$-HE implies the
same limit for fixed trace $\beta$-Hermite ensembles. Consequently, explicit
limit can be given for fixed trace GOE, GUE and GSE. Furthermore, for even
$\beta$, analogous to $\beta$-Hermite ensembles, a multiple integral of the
Konstevich type can be obtained.
