We have tested a rigorous version of the Island Rule for the long-term mean magnitude of the Indonesian Throughflow, in a Bryan-Cox model of the Pacific and Indian Oceans. We assign specific causes, in definite locations, to departures from the original version of the rule. Some of these causes can be tested observationally. If Australasia's northern tip is taken at the equator, then in the original version of the rule the throughflow magnitude can be calculated as follows. (1) Everywhere along the standard Island Rule path from Chile to Australasia's southern tip, via the equatorial Pacific, the long-path gradient ∂P/∂l of depth-integrated steric height (DISH) P is given by the long-path wind stress &tgr;l divided by density &rgr; and gravitational acceleration g. (2) The Indonesian Throughflow is the sum of the geostrophic flow balancing the resulting DISH difference ΔP between Chile and southern Australasia and the northward Ekman transport between Chile and Australasia. (Corrections for the pressure difference across New Zealand; for flows through Bass, Torres, and Bering Straits; for pressure gradients at the sill depth; and for the Pacific-wide vertical transport W through the sill depth are all treated here as effects omitted from the original version, as are all nonlinear and frictional contributions.) In the model, Torres and Bering Straits were closed and Tasmania was submerged, so the flows through minor straits were all zero. Local balance of wind stress by the DISH gradient worked well along the South American coast and along the western coast of Australasia north of 20 ¿S. In particular, no large frictional components developed near the eastern side of the narrow Indonesian gap. Lateral friction and nonlinearity introduced quite large contributions across the equatorial Pacific, though they largely cancelled one another. However, major departures from the balance assumed in the Island Rule (relative to the sill depth, 629 m in our model) occurred along south western and southern Australia, owing to substantial longshore pressure gradients at the sill depth. These longshore pressure gradients appear to drive the model's Leeuwin Undercurrent. There was also a moderate Coriolis contribution from net outflow near Cape Leeuwin that is supplied by upwelling along Australia's south coast. These effects combined to reduce the model's value of ΔP to 34.6 m2, well below the Island Rule value of 60.4 m2. However, the corresponding depth-integrated pressure (DIP) difference was very close to the original value, because pressure gradients at sill depth largely compensated for the reductions in DISH. The pressure difference across New Zealand was not small; it took 1.0 Sv away from the original estimate. Pacific-wide upwelling W through sill depth was -0.5 Sv, and departures from geostrophy across the South Pacific contributed 0.1 Sv. It was also found that the DIP difference calculated around the direct and indirect routes from Chile to Australasia differed by 3.8m2, an indication of the model's inability to precisely close the pressure integral over long, closed paths that follow complex topography. Several of these correction terms cancelled, so that the full model value of the throughflow (9.5 Sv; 1Sv=106 m3 s-1) was not very different from the original value. This was 10.5 Sv for the wind and topography of our model. ¿ 1999 American Geophysical Union