1893.] Elasticity of a Crystal according to Boscovich. 



69 



22. Confining our attention now to O's nearest neighbours, we 

 see that the nine not shown in the diagram are in the middles of 

 squares obtained by producing the lines YO, ZO, XO to equal 

 distances beyond O 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 \ the distance of each of them from 0, we have 

 for the coordinates x t y, z of these twelve points as follows : 



\\ / x x \ 



-2-' -5' "il 



A A \ /A A 



\/ 2 v 2 



/ V 



"TV"?*' 



.... (28). 



23. Suppose now O 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 -ar for all. These yield eight 

 terms to 2 (x^r*), and four to 2 (^/V/r 4 ) ; and the value of each term 

 in these sums is . Thus we find that 



A = Nw, and B = n = 

 Hence and by (24), we see that 



(29). 

 (30). 



