1915-16.] Fa]l of Small Particles through Liquid Columns. 237 
XIV.— Mathematical Note on the Fall of Small Particles 
through Liquid Columns. By Professor C. G. Knott. 
(MS. received March 20, 1916. Read March 20, 1916.) 
The following note presents in a simpler form the essence of Dr Sven 
Oden’s mathematical discussion of the fall of small particles through a 
column of liquid, as given in his paper (immediately preceding) “ On the 
Size of Particles in Deep-sea Deposits.” 
It has the further advantage of solving the problem without taking 
account explicitly or implicitly of Stokes’s law (or modification thereof) 
as to the relation connecting the time of fall with the size, form, and 
density of the particles considered. It is important, I think, to refrain 
as long as possible from making the assumptions involved in such a law 
of fall, and to recognise how far we may carry the investigation before 
introducing these assumptions. 
Imagine, then, a great number of small particles falling through a 
column of water. The terminal velocity and therefore rate of fall 
depend in some unknown way (unless the particles are spheres) upon 
their size and shape. Let them be partitioned off in bundles so that 
their times of fall form an arithmetical progression with small constant 
difference St * 
The “accumulation-curve” for any one of these groups will be a 
straight line whose final ordinate y r corresponds to the time rSt taken by 
the corresponding set of particles to fall through the whole height. 
Particles of shorter time of fall will have already accumulated on the 
bottom. Particles of longer time of fall will have partially accumulated, 
the fraction of y s ( s greater than r) which has. accumulated up to time 
T 
rSt being ~y s . 
s 
Hence the total accumulation up to time rSt is 
p r = 2 + 2 
i p = i 
where nSt is the whole time for the completed accumulation. 
* We can imagine this being done by Maxwell’s demons or other acute intelligences. 
