Summary: Under the assumptions that \(E_{\lambda}^n\) is an \(n\)-dimensional, simply connected Riemannian manifold of constant sectional curvature \(\lambda\) and \(L_{\lambda}^r\) is an \(r\)-dimensional, totally geodesic submanifold of \(E_{\lambda}^n\), the paper investigates the \(q\)-th integral of the mean curvature \(M_{q}^n\) of a convex body \(K^r\) in \(E_{\lambda}^n\) and gives the expression of \(M_{q}^n\) in the terms of \(M_p^r\) where \(M_{p}^r\) is the \(p\)-th integral of the mean curvature of \(K^r\) in \(L_{\lambda}^r\). A result of L. A. Santaló [Rev. Fac. Sci. Univ. Istanbul, Sér. A 21, 189–194 (1957; Zbl 0091.35503)] holds in particular.