If P be the resistance encountered by a solid spherule of radius r, moving through a viscous liquid at the rate x, and if k be the frictional coefficient, then P = 6πkrx. Again, the effective part of the weight of the particle is P´ = ⁴⁄₃πr³ (ρ-ρ´)g, where g is the acceleration of gravity and ρ and ρ´ the density of solid particle and liquid, respectively. In case of uniform motion P = P´. Hence x = 2/9kr² (ρ-ρ´)g ... (1). In any given case of thoroughly triturated material the particles vary in size from a very small to a relatively large value; but by far the greater number approach a certain mean figure and dimension. An example of this condition of things may be formulated. To avoid mathematical entanglement let y = Ax^{³⁄₂}e^{-x²} ... (2) where y is the probable occurrence of the rate of subsidence x. If now the turbidity of the liquid (avoiding optical considerations) be defined as proportional to the mass of solid material particles suspended in unit of volume of liquid, then the degree of turbidity which the given ydx particles add to the liquid is, _caeteris paribus_, proportional to r³ydx, where r is the mean radius. Hence the turbidity, T, at the outset of the experiment (immediately after shaking), is T = T₀∫₀^∞r³ydx = T₀, where equations (1) and (2) have been incorporated. If the plane at a depth d below the surface of the liquid be regarded, then at a time after shaking the residual turbidity is (3) ... T_{d} = T₀∫^{d/t}₀r³ydx = T₀(1 − (1 + (d²/t² × e^{-d²/t²})) The equation describes the observed occurrences fairly well. The phenomena of stratification observed by Brewer are explained by Barus from the above formula: In proportion as the time of subsidence is greater, the tube shows opacity at the bottom, shading off gradually upward, through translucency, into clearness at the top. If, instead of equation (2), there be introduced the condition of a more abrupt maximum, if, in other words, the particles be very nearly of the same size, then subsidence must take place in unbroken column capped by a plane surface which at the time zero coincided with the free surface of the liquid. Again, suppose one-half of the particles of this column differ in some way uniformly from the other half. Then at the outset there are two continuous columns coinciding, or, as it were, interpenetrating throughout their extent. But the rate of subsidence of these two columns is necessarily different, since the particles, each for each, differ in density, radius and frictional qualities, by given fixed amounts. Hence the two surfaces of demarcation at the time zero coincided with the free surface. In general, if there be n groups of particles uniformly distributed, then at the time zero n continuous columns interpenetrate and coincide throughout their extent. At the time t, the free surface will be represented by n consecutive surfaces of demarcation below it, each of which caps a column, the particles of which form a distinct group.