216 Proceedings of Royal Society of Edinburgh. [sess. 
chemical molecules, and in many cases are composed of groups of 
the chemical molecules. The crystalline molecules, however 
constituted, are, in crystals of the cubic class, probably arranged 
either in simple cubic, or double cubic, or in simple equilateral, or 
double equilateral, order. 
§ 19. It will be an interesting further development of the 
molecular theory to find some illustrative cases of chemical 
compound molecules (that is to say, groups of atoms presenting 
different laws of force, whether between two atoms of the same 
kind or between atoms of different kinds), which are, and others 
which are not, in stable equilibrium at some density or densities 
of equilateral assemblage. In this last class of cases the molecules 
make up crystals not of the cubic class. This certainly can be 
arranged for by compound molecules with law of force between 
any two atoms fulfilling the condition of § 13; and it can be done 
even for a monatomic homogeneous assemblage very easily, if we 
leave the simplicity of § 13 in our assumption as to law of 
force. 
§ 20. The mathematical theory wants development in respect 
to the conditions for stability. If, with the constraining guidance 
of § 16, w is either a maximum or a minimum, there is equilibrium 
with or without the guidance. For iv a maximum the equilibrium 
is stable with the guidance ; but may be stable or unstable without 
the guidance. A criterion of stability which will answer this 
last question is much wanted ; and it seems to me that though the 
number of atoms is quasi infinite the wanted criterion may be finite 
in every case in which the number of atoms exerting force on any 
one atom is finite. To find it generally for the equilibrium of 
any homogeneous assemblage of homogeneous groups, each of a 
finite number of atoms, is a worthy object for mathematical con- 
sideration. Its difficulty and complexity is illustrated in §§ 21, 
22 for the particularly simple case of similar atoms arranged in 
simple cubic order; and in §§ 23-29 for a still simpler case. 
§ 21. Consider a group of eight particles at the eight corners of 
a cube (edge X) mutually acting on one another with forces all 
varying according to the same law of distance. Let the magni- 
tudes of the forces be such that there is equilibrium ; and in the 
first place let the law of variation of the forces be such that the 
