Previous work has established that the linear correlation of the low-altitude particle cusp latitude with the southward component of the IMF is about 0.70. Several possibly better candidate functions for determining the coupling between the magnetosphere and the solar wind have been advanced, but none have been evaluated in terms of the cusp, which is a site of direct solar wind--magnetosphere interaction. On the basis of 11 years of DMSP satellite particle data from 1984--1994 (with verification from the subsequent 11 years, 1995--2005), we find that the best solar wind--magnetosphere coupling function involves electric field dimensions, such as the half wave rectifier (vBs) and the Kan-Lee electric field (EKL = vBTsin2($theta$c/2), where $theta$c is the IMF clock angle). Both the half wave rectifier (r = 0.77) and the Kan-Lee (r = 0.78) functions have a linear correlation with cusp latitude which is noticeably better than the Bz function used in previous work, and also better than the $varepsilon$ parameter ($varepsilon$ = vB2sin4($theta$c/2)). However, the best correlation is with a function whose clock angle dependence is intermediate between the pure half wave rectifier (which implies no merging for Bz > 0) and the Kan-Lee function. Namely, EWAV = vBTsin4($theta$c/2) correlates with cusp latitude at the r = 0.81 level. This latter clock angle dependence has been previously suggested at various times by J. R. Wygant, by S.-I. Akasofu, and by V. M. Vasyliunas. The improved result holds for both the equatorward and poleward edge of the cusp, and regardless of how the IMF is propagated. Earlier work on cross polar cap potentials and on nightside auroral luminosity also favored the EWAV function, which in combination with our findings suggests a widely applicable result. Dayside merging is thus clearly not purely component driven, as the half wave rectifier formula implies. These results also suggest, albeit less convincingly, that merging shuts down for increasingly northward IMF more rapidly than the Kan-Lee electric field implies.