424 Lord Kelvin on the Elasticity 



of the near faces of the principal cnbe shown in the diagram ; 

 and three of its six next-nearest neighbours (their distances 

 Xx/2), being at X, Y, Z, the corners of the cube nearest to 

 it ; and, at other corners of the cube, three other neighbours, 

 K, L, M, which are next-next-next-nearest (their distances 2X) . 

 The points in the middles of the three remote sides of the cube, 

 not seen in the diagram, are next-next-nearest neighbours of 

 (their distances XV 3). 



§ 22. Confining our attention now to O's nearest neigh- 

 bours, we see that the nine not shown in the diagram are in 

 the middles of squares obtained by producing the lines Y 0, 

 Z 0, X to equal distances beyond and completing the 

 squares on all the pairs of lines so obtained. To see this more 

 clearly, imagine eight equal cubes placed together, with faces 

 in contact and each with one corner at 0. The pairs of faces 

 in contact are four squares in each of the three planes cutting 

 one another at right angles through ; and the centres of 

 these twelve squares are the twelve nearest neighbours of 0. 

 If we denote by X the distance of each of them from 0, we 

 have for the coordinates sc, y, z of these twelve points as 

 follows : — 



(«• 7T 7i> («■ - ■ 7T 7i> (»• 7? " 7*> (»> "T? " T> 

 (7V "• 72} (" 72' "• 7s) (7? °' " 75) ( - 75' "• ~ 75) 



§ 23. Suppose now to experience force only from its 

 twelve nearest neighbours ; the summations 2 of § 18 (23) 

 will include just these twelve points with equal values of ot 

 for all. These yield eight terms to 2(^/r 4 ), and four to 

 %(y 2 z 2 /r 4 ) ; and the value of each term in these sums is J. 

 Thus we find that 



A=]fe, and B = n=p^7 .... (29). 

 Hence and by (24), we see that 



n x =\n (30). 



Thus we have the remarkable result that, relatively to the 

 principal cube, the diagonal rigidity is half the facial rigidity 

 when each point experiences force only from its twelve nearest 

 neighbours. This proposition was announced without proof 

 in § (28) of " Molecular Constitution of Matter"*. 

 * 'Math, and Phys. Papers,' vol. iii. p. 403. 



