256 Prof. J. J. Thomson on the 
atom by this ring structure are analogous in many respects 
to those possessed by the atoms of the chemical elements, and 
that in particular the properties of the atom will depend upon 
its atomic weight in a way very analogous to that expressed 
by the periodic law. 
Let us suppose, then, that we have N corpuscles each 
carrying a charge e of negative electricity, placed in a sphere 
of positive electrification, the whole charge in the sphere 
being equal to Ne; let us find the distribution of the cor- 
puscles when they are arranged in what we may consider to 
be the simplest way, 2.e. when the number of rings is a 
minimum, so that in each ring there are as nearly as possible 
as many corpuscles as it is possible for the corpuscles inside 
to hold in equilibrium. Let us suppose that the number of 
internal corpuscles required to make the equilibrium of a 
ring of 2 corpuscles stable is f(n). The value of f(m) for a 
series of values of n is given in the table on page 254; in 
that table f(n) is denoted by p. The number of corpuscles 
in the outer ring 7, will then be determined by the condition 
that N—n,, the number of corpuscles inside, must be just 
sufficient to keep the ring of mn, corpuscles in equilibrium, 
1. @., n, will be determined by the equation. 
Nisa f(y ye! reas 2 a 
If the value of x, got from this equation is not an integer 
we must take the integral part of the value. 
To get nz, the number of corpuscles in the second ring, we 
notice that there must be N —n,—n, corpuscles inside; hence 
Ny 1s given by the equation 
N = 1 —1=f (ng). 
Similarly, 23, 2, ..., the number of corpuscles in the 3rd, 
Ath, &c. rings reckoned from the outside, are given by 
N—17,—7—n3=f (ns), 
N—1 —n,—7n3—1g=/ (n4). 
These equations can be solved very rapidly by a graphical 
method. Draw the graph whose abscissa =/(n) and whose 
ordinate is n. The values of /(n) for a series of values of n 
are given on page 254; from these values the curve fig. 1 
has been constructed. 
To find howa number of corpuscles equal to N will arrange 
themselves, measure off on the axis of abscissze a distance 
from O equal to N. Let OP be this distance, through P 
draw PQ inclined at,an angle of 135° to the horizontal axis, 
