  Let $(\mathrm{Spec }R, \frak m)$ be a rational double point defined over an
algebraically closed field $k$ of characteristic $p\geq 0$. We evaluate further
the dimensions of the local cohomology groups which were treated by Wahl in
1975 as vanishing theorem C (resp. D) under the assumption that $p$ is a very
good prime (resp. good prime) with respect to $(\mathrm{Spec }R, \frak m)$. We
use Artin's classification of rational double points and completely determine
the dimensions $\dim_k H_E^1(S_X)$, $\dim_k H_E^1(S_X\otimes \mathcal O_X(E))$,
supplementing Wahl's theorems. In the proof we construct derivations concretely
which do not lift to the minimal resolution $X\to \mathrm{Spec }R$, as well as
non-trivial equisingular families which inject into a versal deformation of the
rational double point $(\mathrm{Spec }R, \mathfrak m)$.
