  We define invariants $\frak{ds}$ and $\overline{\frak{ds}}$, which are the
maximal and minimal second Betti number divided by $8$ among definite spin
boundings of a homology sphere. The similar invariants $g_8$ and
$\overline{g_8}$ are defined by the maximal (or minimal) product sum of
$E_8$-form of bounding 4-manifolds. We compute these invariants for some
homology spheres. We construct $E_8$-boundings for some of Brieskorn 3-spheres
$\Sigma(2,3,12n+5)$ by handle decomposition. As a by-product of the
construction, some negative classes which consist of addition of several fiber
classes plus one sectional class in $E(1)$ are represented by spheres.
