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We consider ruled and quadric surfaces in the 3-dimensional Euclidean spacewhich are of coordinate finite type with respect to the third fundamental form$III$, i.e., their position vector $\boldsymbol{x}$ satisfies the relation$\Delta ^{III}\boldsymbol{x}=\varLambda \boldsymbol{x}$ where $\varLambda $ isa square matrix of order 3. We show that helicoids and spheres are the onlysurfaces in $E^3$ satisfying the preceding relation.

Ruled and quadric surfaces in the 3-dimensional Euclidean space satisfying $\Delta ^{III}\boldsymbol{x} = \varLambda \boldsymbol{x}$