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Proceedings of the Royal Society of Edinburgh. [Sess. 
liquid, so that the multiplier of r 2 is a constant dependent on the nature 
of the liquid and of the sphere. Consequently, by measuring v in some way 
or other we can calculate the value of r, i.e. the size of the particle. 
Tiiis purely theoretical law has been experimentally investigated by 
H. S. Allen,* H. D. Arnold, f I. NordlundJ and others. For particles of 
dimensions exceeding a certain limit it has been found that the actual 
velocity is higher than its theoretical value. Thus Allen has found that 
quartz spherules in water fall strictly according to the law, provided their 
radius does not exceed 85//, ( = 0'085 mm.). Consequently, if the experi- 
ments should give a value for the radius higher than 85//, we can only say 
that the true value does not exceed that found from the experiments. 
We will now consider the case of a particle which is not spherical. It 
is then rather difficult to define its dimensions, especially if its shape is very 
irregular. In order to evade this difficulty, it is convenient to define a new 
quantity, the “ effective radius ” i.e. the radius of a perfect sphere of the 
same material which sinks at the same average rate as the particle in 
question, the latter being supposed to retain during its fall a certain 
orientation with respect to its line of motion. The case of a particle shaped 
like an ellipsoid of rotation may be given here, the effective radii for a 
fall parallel to either axis, a and 6, being : § 
That the effective radius must vary according to the orientation of the 
particle is especially obvious in the case of particles shaped like discs 
or rods. One would perhaps for this reason be inclined to consider any 
attempt to analyse sediments as hopeless. This conclusion would be true 
for extremely small samples consisting of a very limited number of particles, 
or for a suspension of small depth where the total distance through which 
each particle falls is very short. The present case is, however, quite different, 
as the number of particles of a certain size present in the sample is 
extremely large, so that we are not measuring the effective radius of any 
single particle, but are determining the mean effective radius of the 
* “ The Motion of a Sphere in a Viscous Fluid,” Phil. Magazine (5), 1, 323-338 (1900). 
f “ Limitations imposed by Slip and Inertia Terms upon Stokes’s Law for the Motion of 
Spheres through Liquids,” Phil. Magazine (6), xxii, 755-775 (1911). 
i “Ueber die Gtiltigkeit des Stokes’schen Gesetzes, etc.,” Arkiv f. Matematik , etc., edited 
by K. Svenska Vet. Akad. i Stockholm, ix, Nr. 13 (1913). 
§ Viz. The Svedberg, “ Ueber die Gestalt der Molekule,” II, Arkiv f. Kemi, etc., utg. av 
K. Svenska Vet. Akad. i Stockholm, v, Nr. 11 (1914). 
