416 . Lord Kelvin on the Elasticity 



crystalline molecules, in the line joining the two points. The 

 very simplest Boscovichian idea of a crystal is a homogeneous 

 group of single points. The next simplest idea is a homo- 

 geneous group of double points. 



§ 5. In the present communication I demonstrate that, if 

 we take the very simplest Boscovichian idea of a crystal, a 

 homogeneous group of single points, we find essentially six 

 relations between the twenty-one coefficients in the quadratic 

 function expressing w, whether in terms of s 1? . . . , s 6 or of 

 />!,..., p 6 . These six relations are such that infinite resist- 

 ance to change of bulk involves infinite rigidity. In the 

 particular case of an equilateral* homogeneous assemblage 

 with such a law of force as to give equal rigidities for all 

 directions of shearing, these six relations give Sk=5n, which 

 is the relation found by Navier and Poisson in their Bosco- 

 vichian theory for isotropic elasticity in a solid. This relation 

 was shown by Stokes to be violated by many real homogeneous 

 isotropic substances, such, for example, as jelly and india- 

 rubber, which oppose so great resistance to compression and so 

 small resistance to change of shape that we may, with but little 

 practical error, consider them as incompressible elastic solids. 



§6.1 next demonstrate that if we take the next simplest 

 Boscovichian idea for a crystal, a homogeneous group of 

 double points, we can assign very simple laws of variation of 

 the forces between the points which shall give any arbitrarily 

 assigned value to each of the twenty-one coefficients in either 

 of the quadratic expressions for w. 



§ 7. I consider particularly the problem of assigning such 

 values to the twenty-one coefficients of either of the quadratic 

 formulas as shall render the solid incompressible. This is 

 most easily done by taking w as a quadratic function of 

 Pu • • • jPgi an( ^- D y taking one of these generalized stress-com- 

 ponents, say p G , as uniform positive or negative pressure in all 

 directions. This makes s 6 uniform compression or extension 

 in all directions, and makes s ly . . . , s 5 five distortional com- 

 ponents with no change of bulk. The condition that the solid 

 shall be incompressible is then simply that the coefficients of 

 the six terms involving p 6 are each of them zero. Thus, the 

 expression for w becomes merely a quadratic function of the 

 five distortional stress-components, p^ . . . , p 5 , with fifteen 

 independent coefficients ; and equations (3) of § 3 above 



* That is to say, an assemblage in which the lines from any point to 

 three neighbours nearest to it and to one another are inclined at 60° to 

 one another : and these neighbours are at equal distances from it. This 

 implies that each point has twelve equidistant nearest neighbours around 

 it, and that any tetrahedron of four nearest neighbours has for its four 

 faces four equal equilateral triangles. 



