It is shown that, if $C$ is a closed curve of a projective $S_{r}$$(r \ge 2)$ and $\Gamma$ denotes any other closed curve lying on the developable surface $\Sigma$ circumscribed to $C$, it is possible to attach to $C$ and $\Gamma$ a projective integral invariant, $\{ C ; \Gamma \}$, having a simple metrical definition (given in n. 3, Cor. I). Moreover, it is proved (Theor. VII and Cor. II) that this invariant $\{ C ; \Gamma \}$ vanishes whenever $\Gamma$ is semialgebraic (i.e., obtainable as the intersection of $\Sigma$ with an algebraic primal of $S_{r}$) and that, if $r \ge 3$, $\{ C ; \Gamma \}$ remains unchanged when $C$ and $\Gamma$ are substituted by their projections from an arbitrary $S_{h}$ on an $S_{r-h-1}$ of $S_{r}$ skew to $S_{h}$$0 \le h \le r-3$. Further similar results are previously obtained, by using certain preliminary simple properties (cf. Theorems I and II), in connection with $C$, $\Gamma$ and a third closed curve $\Delta$ lying on $\Sigma$.