The Hydrion Differentiation Theory of Geotropism 77 
.. 71 
particle is small compared with the critical radius , which is 
where rj = viscosity of medium, p = density of medium, and 
v = velocity of particle 1 . Arnold found that the law was valid so 
long as the radius of the particle was less than -6 of the critical radius 
(op. cit.p. 96); while Millikan working with gases found that Stokes’ 
Law becomes invalid only when the radius of the sphere is com¬ 
parable to “the mean free path” of a gas molecule, and mentions 
(op. cit. p. 98) that the “holes” in the medium or the mean free paths 
are negligibly small “when the drop falls through a liquid.” 
It seems clear, therefore, that although the exponential “rare¬ 
faction law” governs the final state of equilibrium, Stokes’ Law 
must govern the actual falling or “creaming” of the particles. We 
may have to apply a correction, as Perrin (op. cit. p. 34) points 
out, for the recoil of the particles as they accumulate at the bottom 
or the top of the cell, i.e. as they approach their exponential distri¬ 
bution, but this would be a correction for an aberration from the 
basal law (Stokes’) governing all slow movements of spheres through 
a relatively viscous medium under a constant unidirectional force. 
Further, although the smaller “ultra-microscopic granules within 
the narrow confines of a cell only 0-05 mm. in height must always 
be little removed from their limiting distribution,” the larger micro¬ 
scopic or almost ultramicroscopic particles in the cell will be far 
removed from their limiting distribution, even in a cell only 30 p, 
in height, and they will begin to become redistributed in a few 
minutes as observed for Perrin’s gamboge particles. 
As an example of how far such particles may be removed from 
their stable distribution it will be sufficient to point out that Perrin 
(op. cit. p. 43), with particles of about 0-3 /x radius, found a height 
of 30 /x sufficient to lower the concentration of the granules to one 
tenth of its value. In this case 10 /x, not “30 Millimetres,’’ in the 
cell were equivalent to 6 kilometres in the air. 
As an example of what is supposed to occur, we will take milk, 
regarded as an ideal emulsion and containing fats as a disperse 
phase with a protein solution as the continuous phase. The density 
difference in an average good sample of milk is 0*17, which is smaller 
than the density difference in Perrin’s experiments, as Professor 
Blackman suggests it would be in the cell. Viscosity we may 
1 See The Electron, by R. A. Millikan, p. 95. Chicago. 1917. With particles 
moving 1-55^ in 7 minutes in a medium of density 1-03 as calculated below 
the critical radius is about one kilometre. 
