1893.] Elasticity of a Crystal according to Boscovich. 61 



4. Each crystalline molecule in reality certainly experiences forcive 

 from some of its nearest neighbours on two sides, and probably also 

 from next nearest neighbours and others. Whatever the mutual 

 forcive between two mutually acting crystalline molecules is in 

 reality, and however it is produced, whether by continuous pressure 

 in some medium, or by action at a distance, we may ideally reduce it, 

 according to elementary statical principles, to two forces, or to one 

 single force and a couple in a plane perpendicular to that force. 

 Boscovich's theory, a purely mathematical idealism, makes each crys- 

 talline molecule a single point, or a group of points, and assumes that 

 there is a mutual force between each point of one crystalline molecule 

 and each point of neighbouring crystalline molecules, in the line join- 

 ing the two points. The very simplest Boscovichian idea of a crystal 

 is a homogeneous group of single points. The next simplest idea is a 

 homogeneous 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^ . . . . , s 6 or of p lt . . . . , p 9 . These six relations 

 are such that infinite resistance 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 3k = 5'?&, which is the 

 relation found by Navier and Poisson in their Boscovichian 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 resist- 

 ance 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. I next demonstrate that if we take the next simplest Bosco- 

 vichian 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. 1 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 p^ . . . . , _p 6 an( l by taking one of these 



* 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 anothar ; 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. 



