  Let $C({\bf n})$ be a complete intersection monomial curve in the
4-dimensional affine space. In this paper we study the complete intersection
property of the monomial curve $C({\bf n}+w{\bf v})$, where $w>0$ is an integer
and ${\bf v} \in \mathbb{N}^{4}$. Also we investigate the Cohen-Macaulayness of
the tangent cone of $C({\bf n}+w{\bf v})$.
