152 Lord Kelvin 



on 



our present purpose, shows for this particular curve three of 

 the solutions of equation (8). (There are obviously several 

 other solutions.) In two of the solutions, respectively, A , A', 

 and A , A." ', are consecutive atoms at distances at which the 

 force between them is zero. These are configurations of 

 equilibrium, because A B, the extreme distance at which 

 there is mutual action, is less than twice A A', and less than 

 twice A Q A! f . In the other of the solutions shown, A , A l5 A 2 , 

 A 3 , A 4 , A 5 , A 6 are seven equidistant consecutive atoms of an 

 infinite row in equilibrium in which A 5 is within range of 

 the force of A , and A 6 is beyond it. The algebraic sum of the 

 ordinates with their proper multipliers is zero, and so the 

 diagram represents a solution of equation (9). 



§ 25. In the general linear problem to find whether the 

 equilibrium is stable or not for equal consecutive distances, a, 

 let (as in § 4) <p(D) be the work required to increase the 

 distance between two atoms from D to cc . Suppose now the 

 atoms to be displaced from equal distances, a, to consecutive 

 unequal distances — 



.... a + w._ 2 , a-f M ._ 15 a + u v a + u i+l , a + u i+2 , . (10). 



The equilibrium will be stable or unstable according as the 

 work required to produce this displacement is, or is not, positive 

 for all infinitely small values of ... . u i _ 1 , u v u i+1 , .... 



Its amount is W —W : w T here W denotes the total amount 

 of work required to separate all the atoms from the con- 

 figuration (10) to infinite mutual distances. 

 According to § 2 above W is given by 



W=i( . . . + Wi _ 1+Wi + W . +1 + ....). (11); 



where 



= $(a + u { ) -f $ (2a 4- u i _ 1 + u.) + <j> (3a -f u { _ 2 + w._ x + wj + . . , 



+ 4>(a + u i+1 )+4){2a + u i+1 +w i+2 ) +<j>(3a + u i+1 +u i+2 +u i+3 ) +. . . 



+ • (12> 



Expanding each term by Taylor's theorem as far as terms of 

 the second order, and remarking that the sum of terms of the 

 first order is zero for equilibrium * at equal distances, a, and 



* It is interesting and instructive to verify this analytically by selecting 

 all the terms in W which contain zi i} and thus finding -^— . This equated 



i 



to zero, for zero values of . . . Wj_i, w,- , u i+1 , . . . gives equation (9) of 

 the text. 



