Before revision: It follows from the sectioning procedure that each principal semiaxis of the ellipsoid is generally not the refractive index for propagation in the direction of that semiaxis, but rather the refractive index for wavefronts tangential to that direction, with the vector parallel to that direction, propagating perpendicular to that direction.
After revision: It follows from the sectioning procedure that each principal semiaxis of the ellipsoid is generally not the refractive index for propagation in the direction of that semiaxis, but rather the refractive index for wavefronts orthogonal to that direction, with the vector parallel to that direction, propagating perpendicular to that direction.