The stability of a periodic magnetic island structure using the ideal magnetohydrodynamic equations has been investigated. An instability is found which describes the tendency toward coalescence of parallel currents in the neighboring islands. It is expected that this instability will proceed at a fast hydromagnetic rate as long as the forces driving the instability can overcome the stabilizing forces due to the compression of the magnetic field between the islands. Beyond that phase, resistivity is expected to dominate the tendency toward island coalescence. Island coalescence of this kind can explain why in the observation of tearing mode instabilities in tokamaks, only the modes with minimum values of m and n are seen.