Molecular Dynamics of a Crystal. 119 



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 mona- 

 tomic 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 w 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 

 consideration. 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 V) mutually acting on one another with forces 

 all varying according to the same law of distance. Let the 

 magnitudes 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 equilibrium is stable. Build up now a quasi 

 infinite number of such cubes with coincident corners to form 

 one large cube or a crystal of any other shape. Join ideally, 

 to make one atom, each set of eight particles in contact which 

 we find in this structure. The whole system is in stable 

 equilibrium. The four forces in each set of four coincident 

 edges of the primitive cubes become one force equal to the 

 force between atom and atom at distance \. The two forces 

 in either diagonal of the coincident square faces of two cubes 

 in contact make one force equal to the force between atoms 

 at distance X\ / 2. The single force in each body-diagonal of 

 any one of the cubes is the force between atom and atom at 

 distance \*/3. The three moduluses of elasticity (compressi- 

 bility-modulus, modulus with reference to change of angles 

 of the square faces, and modulus with reference to change of 



