In parts I and II of this series (6, 7), the performance of dynamically loaded sleeve bearings was calculated neglecting the transverse flow in the oil film, i.e., the bearing was assumed to be wide compared to its diameter, and then side-leakage factors based upon constant-load calculations were applied. In the present paper the performance is calculated for bearings assumed to be so narrow that film flow due to pressure in the direction of motion can be neglected. In this case the dependence upon bearing width appears explicitly and no side-leakage factors are required. Conclusions about the performance of the bearing under various simple types of loading are drawn. The two solutions for the limiting cases of very wide and very narrow bearings are compared and found to be qualitatively similar. From this it is concluded that bearings of arbitrary width should show the same qualitative behavior. Finally, the present solution is compared with exact calculations for certain narrow bearings, and the error involved in the present approximation is thereby determined.