Solid and liquid particulate materials that are ejected byhydrothermal or volcanic eruptions are subsequentlydispersed by atmospheric wind currents while falling undergravity to the Earth's surface. Particle sizes are not uniformand may indeed change during flight owing to coalescenceand/or fragmentation. Wind conditions (speed, directionand turbulence) may also change with elevation (and withtime).In an attempt to determine the most important physicalfactors, a quantitative model that reflects the aboveinfluences on particle dispersal is outlined. Analyticsolutions to the mathematical model are sought whereverpossible, since an analysis of the sensitivity of the predicteddistributions to the numerous parameters involved is thenmore readily undertaken. Numerical simulations have alsobeen made. Some example calculation results arecompared with measured eruption deposit patternspublished in the geological literature; these comparisonsindicate that the proposed model reflects variouscharacteristics of the measured data well.