Attention is confined to roll-cell development and roll-cell interaction appropriate to one horizontal dimension larger than either the other horizontal dimension or the depth. At simple eigenvalues Mc the roll-cell amplitude and transport fields can be obtained. Near those aspect ratios corresponding to double eigenvalues Mc, where two roll-cell states of linear theory areequallylikely, thenonlinear theory predicts sequences of transitions from one steady convective state to another as the Marangoni number is increased. Direct comparisons are made of the results here with those of the previous paper for Marangoni convection in circular cylinders. Time-periodic convection is possible in certain cases.