Molecular Dynamics of a Crystal. 



155 



remove the tie between A 3 and A 4 and find, still by a single 

 equation, the altered distance a + ^3, and so on till we find 

 ^i-j or v v s or ^ small enough to be negligible. Thus found, 

 i«i, v v 2 , 1^3, . . . . \X i give a first approximation to the devi- 

 ations from equality of distance for complete equilibrium. 

 Repeat the process of removing the ties in order and replacing 

 each one by the altered length as in the first set of approxi- 

 mations, and we find a second set 2 #i, 2^27 2 x z .... Gro on 

 similarly to a third, fourth, Hfth, sixth .... approximation 

 till we find no change by a repetition of the process. Thus, 

 by a process essentially convergent if the equilibrium with 

 which we started is stable, we find the deviations from equality 

 of consecutive distances required for equilibrium when the 

 system is left free in the neighbourhood of each end, and all 

 through the row (except always the constraint to remain in a 

 straight line). By this proceeding applied to the curve of 

 fig. 7 and the case of equilibrium <x = *680, the following 

 successive approximations were found : — 



1st Approximation 



2nd 



3rd 



4th 



oth 



6th 



7th 



8th 



x., 



+ •018- -009 +004 



j +-02() --014 +007 



+•031 I- -018 +-009 



+ •034 - 1)20 +-011 



+ •036 --022 +-012 



+ •037 --023 +-013 

 I+-038 --024 

 +•039 



*4 



*3 



a'e 



x 7 



-•002 



+•001 



-001 



■ 



•000 



-•003 



+•002 







-005 



+•003 







-006 









-007 









Tims our final solution, with a = '680, is 



= -r-039, x 2 - -'024, #g= +-013, # 4 = — -007, x- y 

 .i' 6 =--001, # 7 = -000. 



+ -003 r 



§ 29. It is exceedingly interesting to remark that the 

 deviations of the successive distances from a are alternately 

 positive and negative, and that they only become less than 

 one-seventh per cent, of a for the distance between A 7 and A 8 . 

 Thus, if we agree to neglect anything less than one-seventh 

 per cent, in the distance between atom and atom, the influential 

 distance from either end is 7a, although the mutual force 

 between atom and atom is null at all distances exceeding 2*2a. 



§ 30. If, instead of /(D) denoting the force between two 

 atoms in a rectilinear row, it denotes the mutual force between 

 two parallel plane nets in a Bravais homogeneous assemblage 



