tures and n respectively, dim V1 = m> 1 and dim V2 =n-m ?1. Mr n-mn In the latter part of this statement, the second fundamental form A has two eigenvalues of multiplicities dim VI and dim V2. The main purpose of the present paper is to investigate the converse problem for minimal hypersurfaces in Sn+1. The author will prove a local theorem on the integrability of the distributions of the spaces of principal vectors of a hypersurface in a Riemannian manifold of constant curvature (Theorem 2). Making use of it, he will investigate minimal hypersurfaces such that the multiplicities of principal curvatures are constant. He will prove that if the number of principal curvatures is two and the multiplicities of them are at least two for a minimal hypersurface of this kind in Sn+', then it is