1901 — 2.] Lord Kelvin on Molecular Dynamics of a Crystal. 221 
where 2 denotes summation for all values of i, except those corre- 
sponding to the small numbers of atoms (§§ 28, 29 below) within 
influential distances of the two ends of the row. 
§ 26. Hence the equilibrium is stable if f'(a), f'(2a), f'(3a), etc., 
are all positive ; but it can be stable with some of them negative. 
Thus, according to the Boscovich diagram, a condition ensuring 
stability is that the position of each atom be on an up-slope of the 
curve showing attractions at increasing distances. We see that 
each of the atoms in each of our three equilibriums for fig. 6 fulfils 
this condition. 
§ 27. Fig. 7 shows a simple Boscovich curve drawn arbitrarily 
to fulfil the condition of § 1 3 above, and with the further 
simplification for our present purpose, of limiting the sphere of 
influence so as not to extend beyond the next-nearest neighbours 
in a row of equidistant particles in equilibrium, with repulsions 
between nearests and attractions between next-nearests. The 
distance, a, between nearests is determined by 
f(a) +:2/(2a) = 0 (14), 
being what (9) of § 24 becomes when there is no mutual force except 
between nearests and next-nearests. There is obviously one stable 
solution of this equation in which one atom is at the zero of the scale 
of abscissas (not shown in the diagram) and its nearest neighbour 
on the right is at A, the point of zero force with attraction for 
greater distances and repulsion for less distances. The only other 
configuration of stable equilibrium is found by solution of (14) ac- 
cording to the plan described in § 24, which gives a= '680. It is. 
shown on fig. 7 by A*, A m , as consecutive atoms in the row. 
§ 28. Consider now the equilibrium in the neighbourhood of 
either end of a rectilinear row of a very large number of atoms 
which, beyond influential distance from either end, are at equal 
consecutive distances a satisfying § 27 (14). We shall take for 
simplicity the case of equilibrium in which there is no extraneous 
force applied to any of the atoms, and no mutual force between 
any two atoms except the positive or negative attraction ^(D). 
But suppose first that ties or struts are placed between consecutive 
atoms near each end of the row so as to keep all their consecutive 
distances exactl} T equal to a. For brevity we shall call them ties, 
