In 1870, E.Schroder showed that the convergence of the Newton process of successive approximations to a multiple solution of a scalar equation was geometric in character, and that quadratic convergence could be restored by multiplying the ordinary corrections by a constant. Here, this result is extended to finite systems, and it is shown that there exist various subspaces of the given space in which the convergence is geometric with a rate characteristic of the given subspace. Quadratic convergence may be restored by applying a given fixed linear operator to the ordinary corrections. The conditions under which these results apply to equations in infinite-dimensional Banach spaces are given. Numerical examples involving scalar equations and a simple 2 × 2 system are presented.