Conditions on a Surface $F^2\subset E^n$ to lie in $E^4$

We consider a surface $F^2$ in $E^n$ with a non-degenerate ellipse of normal curvature whose plane passes through the corresponding surface point. The definition of three types of points is given in dependence of the position of the point relatively to the ellipse. If in the domain $D\subset F^2$ all the points are of the same type, then the domain $D$ is said also to be of this type. This classification of points and domains is linked with the classification of partial differential equations of the second order. The theorems on the surface to lie in $E^4$ are proved under the fulfilment of certain boundary conditions. Some examples of the surfaces are constructed to show that the boundary conditions of the theorems are essential.