For a strong, stationary double layer to be maintained, the electron flux PHI(e) and the ion flux PHI(i) must satisfy the Langmuir condition PHI(e)/PHI(i) = square-root m(i)/m(e). However, this condition alone does not determine whether or not the double layer will remain in stable equilibrium at a given position. A new stability condition is obtained and the results of several laboratory double layer experiments are examined using this condition.