Plucker-Clebsch formula in higher dimension

Let $S\subset {\text{Ps}}^r$ ($r\ge 5$) be a nondegenerate, irreducible, smooth, complex, projective surface of degree $d$. Let ${\delta }_S$ be the number of double points of a general projection of $S$ to ${\text{Ps}}^4$. In the present paper we prove that ${\delta }_S\le \left(\genfrac{}{}{0px}{}{d-2}{2}\right)$, with equality if and only if $S$ is a rational scroll. Extensions to higher dimensions are discussed.