Molecular Dynamics of a Crystal. 153 



putting <f>"(D) = -/'(D), we find 

 W -W=i${f'(a){u* + u* i+1 ) 



+f(-2a)[(u i _ l +u i y+(u i+l +u i+2 yi 



+/ , (3«)[K_ 2 + u i _ 1 +utf + (ii l+l +u i+2 +« i+ ,) 2 ] 



-f etc. etc. etc. etc. \ (13) 



where 2 denotes summation for all values of i, except those 

 corresponding to the small numbers o£ atoms (§§ 28, 29 

 below) within influential distances o£ the two ends of the row. 



§ 26. Hence the equilibrium is stable if f (a), /' (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. YY'e 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 arbi- 

 trarily to fulfil the condition of §13 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, «, between nearests is deter- 

 mined by 



/(«) + 2/(2«)=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) according to the plan 

 described in 5 24. which gives a = '680. It is shown on fig. 7 

 by A t -, A l+1 , 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 



