  Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this
article we will give an alternative and elementary proof of Lelli Chiesa's
Theorem in the case of $r= 2$. More precisely we will prove that that under
certain condition the second co-ordinate of the gonality sequence is constant
along the smooth curves in the linear system $|L|$. Using Lelli Chiesa's
theorem for $r \ge 3$ we also extend Lelli Chiesa's Theorem in the case of $r=
2$ in weaker condition.
