Hypersurfaces with constant principal curvatures in Euclidean space $V^{n+1}$

Hypersurfaces in $E^{n+1}$ for which a thin fan is found are considered. It is shown that it exists only for hypersurfaces in $E^{n+1}$ with constant or proportional principal curvatures that differ from each other. The conditions for the existence of hypersurfaces in the Euclidean space $V^{n+1}$, whose main curvatures are constant (assuming that all the main curvatures are different from each other), are clarified.