This paper deals with properties of homotopies for finding roots of smooth mappings from \(R^ n\) to \(R^ n\). The homotopy considered is of the form: \(\Phi_ a(x,\lambda)=(1-\lambda)(x-a)+\lambda f(x)\), where x is a n-vector, f(x) is an n-vector and f is analytic in the real sense, and points \(x_ 0\) with \(f(x_ 0)=0\) are desired. Two new results are proven in the paper. The first states that, if \((1,x_ 0)\) is a limit point of the curve (x(t),\(\lambda\) (t)) which passes through (a,0), then there is a neighborhood of \((1,x_ 0)\) such that \((\Phi_ a)^{-1}(0)\) consists of finitely many parametrizable curves which can be represented by convergent fractional power series. This result is a significant statement about the behavior of homotopies when they converge to a root at which the Jacobian of f is singular. The second theorem states that the set of a for which the Jacobian of \(\Phi_ a\) is of full rank upon the interior of that component of \((\Phi_ a)^{-1}(0)\) which goes through (a,0) is open and dense. This is a stronger statement than statements based on Sard's theorem, which generally say that the set of a for which the Jacobian is rank-deficient has measure 0. The paper is carefully written.