Submanifolds of finite type were introduced by B. -Y. Chen([2]) about ten years ago. Many works have been done in characterizing or classifying submanifolds in Euclidean space with this notion. On the other hand, the Gauss map is one of the most useful tools to study submanifolds of Euclidean space. In a work of finite type theory, B,-Y. Chen and pp. Piccini ([3]) studied compact submanifolds with finite type Gauss map. In general, non-compact manifolds are usually difficult to be handled. In this article we are going to deal with a part of the following most general question: What kind of surfaces in 3-dimensional Euclidean space have 1-type Gauss mapulcorner