The behavior of a nonrelativistic many-electron wave function when one electron approaches either a nucleus or another electron is well known to be governed by a cusp condition. This is a consequence of the Coulomb singularity in the potential and provides a specific relation between the wave function and its first derivative at the point of coalescence. It is shown here that the coalescence behavior also uniquely determines the third derivative of the spherically averaged wave function in terms of the lower derivatives. The new relation is valid for any atom, molecule, or electron gas in any smooth external field. Further extensions are also discussed for electron-electron coalescence in electron gas systems and for electron-nucleus coalescence in multiconfigurational self-consistent-held wave functions,