422 Lord Kelvin on the Elasticity 



point ; and with no simplifying assumption as to the law of 

 force, we have in the quadratic for w equal values for the 

 coefficients of fg and \a 2 ; ge and -J6 2 ; ef and Jc 2 ; be and ea ; 

 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, Y, Z parallel to the three pairs of oppo- 

 site 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 



tf 4 y 4 z* i/ 2 ? 2 * 2 x 2 X 2 V 2 



(19) becomes 



ic = ±A(e 2 +f+g 2 ) + B(fg+ge + ef) + ±n(a 2 + b 2 + c 2 ) (22), 

 where A =iN%^ and B=a==iN2*r^ . . (23). 



§ 19. Looking to Thomson and Tait's 'Natural Philo- 

 sophy/ § 695 (7)*, we see that n in our present formula (22) 

 denotes the rigidity-modulus relative to shearings parallel to 

 the planes YOZ/ZOX, X Y ; and that if we denote 

 by Wj the rigidity -modulus relative to shearings parallel to 

 planes through OX, Y, Z, and cutting (OY, OZ), 

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

 the compressibility-modulus, we have 



Wl =i(A-B); k=i{A + 2B) 



}••• ( 24 ); 



and our expression (22), for the elastic energy of the strained 

 solid, becomes 



2ic = (k + ±n 1 Xe 2 +f 2 +g 2 ) + 2(k-2n 1 )(fg + ge + ef) 



+ n{a 2 + b 2 + c 2 ) . . (25). 



§ 20. Using in (24) the equality B = n shown in (23), we 

 find 



3k = 2n 1 + 3n (26). 



This remarkable relation between the two rigidities and the 



* 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. 





