150 Lord Kelvin on 



angles between their diagonals) are all easily found by 

 consideration o£ the dynamics of a single primitive cube, or 

 they may be found by the general method given in " On the 

 Elasticity of a Crystal according to Boscovich " *. (In 

 passing, remark that neither in this nor in other cases is it to 

 be assumed without proof that stability is ensured by positive 

 values of the elasticity moduluses.) 



§ 22. Now while it is obvious that our cubic system is in 

 stable equilibrium if the eight particles constituting a detached 

 primitive cube are in stable equilibrium, it is not obvious 

 without proof that this condition, though sufficient, is neces- 

 sary for the stability of the combined assemblage. It might 

 be that though each primitive cube by itself is unstable, the 

 combined assemblage is stable in virtue of mutual support 

 given by the joinings of eight particles into one at the corners 

 of the cubes which we have put together. 



§ 23. The simplest possible illustration of the stability 

 question of § 20 is presented by the exceedingly interesting 

 problem of the equilibrium of an infinite row of similar 

 particles, free to move only in a straight line. The considera- 

 tion of this linear problem we shall find also useful (§§ 28> 

 29 below) for investigation of the disturbance from homo- 

 geneousness in the neighbourhood of the bounding surface, 

 experienced by a three-dimensional homogeneous assemblage 

 in equilibrium. First let us find a, the distance, or one of 

 the distances, from atom to atom at which the atoms must be 

 placed for equilibrium ; and after that try to find whether 

 the equilibrium is stable or unstable. 



§ 24. Calling /(D) (as in § 4) the attraction between atom 

 and atom at distance D, we have for the sum, P, of attractions 

 between all the atoms on one side of any point in their line, 

 and all" the atoms on the other side, the following finite 

 expression having essentially a finite number of terms, greater 

 the smaller is a : 



/(a) + 2/(2a)+3/(3a) + . . . . =P . . . (8). 



Hence a, for equilibrium with no extraneous force, is given 

 by the functional equation 



/(a)+2/(2a)+3/(3a) + .... =0 . . . (9); 

 which, according to the law of force, may give one or two or 

 any number of values for a : or may even give no value (all 

 roots imaginary) if the force at greatest distance for which 

 there is force at all, is repulsive. The solution or all the 

 solutions of this equation are readily found by calculating 



* Proc. R.S.L., vol. 54, June 8, 1893. 



