206 FLUIDITY AND PLASTICITY Einstein1 and Hatschek2 have both considered theoretically the case of suspensions of spherical particles at low concentrations. They both arrive at the formula H = rn (1 + Jfcfe) or , _ «*l /79N * - r+Tb (72) where 6 is the fraction of solid present by volume and k is a constant for which Hatschek deduced the value of 4.5 and Ein- stein of 1. The formula is hyperbolic in form while the formula obtained from available experimental material is linear. Their curve is concave upward, and if it held for high concentrations the pure solid would have a fluidity of 18 per cent (Hatschek) to 50 per cent (Einstein) of the fluidity of the continuous medium, which is absurd. Hatschek states, "It is obvious that the liquid at the upper pole of each spherical particle moves with a somewhat greater velocity than at the lower pole, which is equivalent to a transla- tory movement of the particles with a velocity equal to half the difference of the two velocities prevailing at the two poles." He thus neglects entirely the rotation of the spheres and assumes that they are moving faster than the stratum of fluid which would pass through their centers. That these two motions are equivalent is at least not self-evident. His formula is ob- tained by the employment of Stokes' formula for a sphere moving through a viscous medium without rotation. The view is commonly held that dilute suspensions have a viscosity which is very little different from that of the dispersion medium, but that as the concentration is increased the viscosity suddenly increases. Thus Ostwald in his Kolloid Chemie states, " The curves and tables show that at certain concentrations there is a very sudden increase in viscosity. For silver and glycogen hydrosols these concentrations are respectively about 3.5 and 30 per cent." If the fluidity is in fact linear as we have indicated is the case, the viscosity curve is hyperbolic. There will naturally be a rather sudden increase in viscosity but it has no significance* The question arises, "Does the glycogen jflmdtfa/-concentration 1 Ann. der Physik., 19, 289 (1906)? 2 Kollmd-Zeitschr., 7, 301 (1901); 8, 34 (1911); Trans. Faraday Soc. (1913).