The authors prove the following two results: Theorem 1. Let \(f: M\to E^ 3\) be a smooth immersion of a smooth surface into \(E^ 3\) with an isolated umbilic at \(p_ 0\in M\) of index \(j\) satisfying \(| j|\geq 1\). Then \(p_ 0\) is a critical point of the mean curvature \(H\) and the Gauss curvature \(K\) and the 3-jet of \(H^ 2-K\) vanishes at \(p_ 0\). Theorem 2. If for a compact orientable surface \(M\) immersed in \(E^ 3\) the critical sets of \(H\) and \(K\) are disjoint, then the number of umbilics is at least \(2|\chi(M)|\), where \(\chi(M)\) is the Euler number of \(M\).