In the study of normal (complex analytic) surface singularities, it is interesting to investigate the invariants. The purpose of this paper is to give a characterization of the vanishing of ${delta}_2$. In [11], we gave characterizations of minimally elliptic singularities and rational triple points in terms of th.. second plurigenera ${delta}_2$ and ${gamma}_2$. In this paper, we also give a characterization of rational triple points in terms of a certain computation sequence. To prove our main theorems, we give two formulae for ${delta}_2$ and ${gamma}_2$ of rational surface singularities.