1893.] Elasticity of a Crystal according to JBoscovich. 67 



f" 



\ 



where r0' (r) + r 2 0" (r) = w .............. (20). 



The terms given explicitly in (19) suffice to show by symmetry all 

 the remaining terms represented by the " &c." 



17. Thus we see that with no limitation whatever to the number 

 of neighbours acting with sensible force on any one point O, and 

 with no simplifying assumption as to the law of force, we have in the 

 quadratic for iv equal values for the coefficients of fg and \ a 2 ; ge and 

 | 6 2 ; ef and \ c 3 ; be and ecu ; ca and eb ; and ab and ec. These equalities 

 constitute the six relations promised for demonstration in 5. 



18. In the particular case of an equilateral assemblage, with 

 axes OX, OY, OZ parallel to the three pairs of opposite edges of a 

 tetrahedron of four nearest neighbours, the coefficients which we 

 have found for all the products except fg, ge, ef clearly vanish ; 

 because in the complete sum for a single homogeneous equilateral 

 assemblage we have +#, +y, +z in the symmetrical terms. Hence, 

 and because for this case 



2 = 2 = 2, and 2*r = 2 , = 2*r . (21), 

 r 4 r 4 r 4 r 4 r 4 r 4 



(19) becomes 



w = A(e*+f- + g*) + B(fg+ge + ef) + n(a* + V + c*) .. (22), 



4 o o 



where A = lN2*r^, and B = rc = iNSsr &' ..... (23). 



19. Looking to Thomson and Taifc's 'Natural Philosophy/ 

 695 (7),* we see that n in our present formula (22) denotes the 

 rigidity-modulus relative to shearings parallel to the planes YOZ, 

 ZOX, XOY ; and that if we denote by n^ the rigidity-modulus rela- 

 tive to shearing parallel to planes through OX, OY, OZ, and cutting 

 (OY, OZ), (OZ, OX), (OX, OY) at angles of 45, and if k denote the 

 compressibility-modulus, we have 



1 

 y ......... (24); 



J 



* This formula is given for the case of a body which is wholly isotropic in respect 

 to elasticity moduluses; but from the investigation in 681, 682 we see that our 

 present formula, (22) or (25), expresses the elastic energy for the case of an elastic 

 solid possessing cubic isotropy with unequal rigidities in respect to these two sets 

 of shearings. **** 



F 2 



