A theoretical model is presented to describe the process whereby plasma is "squeezed out" along interplanetary field lines in the magnetosheath near the magnetopause. This squeezing of interplanetary flux tubes resident in the magnetosheath is essentially due to compressional stresses exerted by the magnetosphere. The net effect of the squeezing is to leave flux tubes near the magnetopause depleted of plasma. An idealization of the squeezing process is formulated using the equations of magnetohydrodynamics and, after linearization, analytical solutions are obtained for the changes in plasma density, velocity and magnetic field due to the squeezing down of a single flux tube. The linear analysis reveals that this flow of plasma out along field lines can be described in terms of a slow magnetosonic wave traveling with an angle of = rr/2 - e with respect to the ambient field, where < e < 1. Numerical solution of the non-linear problem is accomplished by considering just a single thin flux tube; values for the plasma density, velocity and magnetic field are obtained as functions of time. The ratio of the density at t = t to the density at anytime t for a particular point, p()/p(t), is termed the depletion factor and evidently increases in time as the tube becomes depleted. The thickness of the layer of depletion outside the magnetopause is estimated using the results of the thin-flux-tube analysis. We call this region of depletion a depletion layer. Thus the thicknesses of depletion layers of given depletion factors are calculated and, for example, a layer with a factor of 2 will be approximately 75-1 kilometers thick near the subsolar point. From the above, one could expect to see a plasma density gradient in the magnetosheath near the subsolar point corresponding to the slope of a plot of depletion factor versus thickness. Although rapid line-merging and cross-field diffusion could render the squeezing effect unimportant and unobservable in many, but not all, cases, it is suggested that the existence of depletion layers might explain the occasionally encountered discrepancy between the position of the magnetopause as determined by plasma data and magnetometer data from the same satellite. The plasma measurements from the Rice University SID experiment aboard ATS-1 are reviewed in conjunction with magnetometer data for the magnetopause traversal of ATS-1 during January 13 and 14, 1967. It is suggested that the low flux measured by ATS-1 in the magnetosheath just after the boundary traversal can be explained by the existence of a depletion layer in the region of the crossing. A preliminary survey of magnetometer data from other satellites suggests that on occasions depletion layers exist.