228 Proceedings of the Royal Society of Edinburgh. [Sess, 
the properties of which are characteristic of each individual sample. In 
another paper I have also proved that these curves are independent of 
the depth of the liquid, h, and also of the total quantity, Poo, of the 
suspended matter, provided that all Aveights are expressed in percentages 
of the latter value, and that the observed intervals of time are reduced to 
the standard depth of 10 cm. by multiplication by 10 jin. 
I shall now proceed to a closer analysis of the accumulation-curve. 
By q(r) I denote the quantity of particles the effective radii of which 
exceed a certain value r. Expressing q(r) in per cent, of the total 
quantity, P*,, we find that A q per cent, of the particles have an effective 
radius of between r and r + Af, A q being the decrease in the value of q 
which corresponds to a small increase, A r, in the argument. The function 
q(r) obviously decreases from its maximum value of 100 (corresponding 
to the smallest particles in the sample) to 0 (for the largest value of r). 
It is now convenient to define another function, F(r), more susceptible 
to mathematical transformations, by which the quantity of particles itself 
is graphically represented by an area. Taking A r quite small, we have 
Ar= o Ar dr 
(the negative sign is retained because q decreases with growing values 
of r). 
Calling — ^2 = F (r), we can, by integrating the function F(r) between 
r x and r 2 , find the percentage of such particles as have values of the 
effective radius comprised within these limits. 
9l~<l2 = 
dq, 
Wr dT = 
F (r)dr 
( 2 ) 
It remains to be shown how this curve F(r), which may conveniently 
be called the “ distribution-curve,” can be found from the accumulation- 
curve given by the experiments. 
Consider a very small interval of F(r) in the vicinity of r ; then 
F (r)dr will represent the fraction of the particles which has an effective 
radius between r and r + dr. After the lapse of a certain time, t, a part 
<p(r) of this fraction will have fallen to the bottom. 
Then 
= (v)drt (3) 
h 
v being the velocity with which the particles of radius r sink through the 
liquid, and h the height from bottom to surface. 
