74 Elasticity of a Crystal according to Boscovich. [June 15, 



as BOA. All the points C' in the nearest plane below are corners 

 of tetrahedrons chirally similar to OA'B'C' placed downwards ou 

 the triangles oriented as B'OA'. The volume of the tetrahedron 

 OABC is } of the volume of the parallelepiped, of which OA, 

 OB, OC are conterminous edges. Hence the sum of the volumes 

 of all the upward tetrahedrons having their bases in one plane is of 

 the volume of the space between large areas of these planes : and, 

 therefore, the sum of all the chirally similar tetrahedrons, such as 

 OABC, is A of the whole volume of the assemblage through any 

 larger space. Hence any homogeneous strain of the assemblage 

 which does not alter the volume of the tetrahedrons does not alter 

 the volume of the solid. Let tie-struts OQ, AQ, BQ, CQ be 

 placed between any point Q within the tetrahedron and its four 

 oorners, and let these tie-struts be mechanically jointed together 

 at Q, so that they may either push or pull at this point. This is 

 merely a mechanical way of stating the Boscovichian idea of a second 

 homogeneous assemblage, equal and similarly oriented to the first 

 assemblage and placed with one of its points at Q, and the others in 

 the other corresponding positions relatively to the primary assem- 

 blage. When it is done for all the tetrahedrons chirally similar to 

 OABC, we find four tie-strut ends at every point 0, or A, or B, or 

 C, for example, of the primary assemblage. Let each set of these 

 four ends be mechanically jointed together, so as to allow either 

 push or pull. A model of the curious structure thus formed was 

 shown at the conversazione of the Royal Society of June 7, 1893. 

 It is for three dimensions of space what ordinary hexagonal netting 

 is in a plane. 



34. Having thus constructed our model, alter its shape until we 

 find its volume a maximum. This brings the tetrahedron, OABC, to 

 be of the special kind defined in 30. Suppose for the present the 

 tie-struts to be absolutely resistant against push and pull, that is to 

 say, to be each of constant length. This secures that the volume of 

 the whole assemblage is unaltered by any infinitesimal change of shape 

 possible to it ; so that we have, in fact, the skeleton of an incompres- 

 sible and inextensible solid.* Let now any forces whatever, subject 

 to the law of uniformity in the assemblage, a,ct between the points of 

 our primary assemblage : and, if we please, also between all the 

 points of our second assemblage ; and between all the points of the two 

 assemblages. Let these forces fulfil the conditions of equilibrium ; of 

 which the principle is described in 16 and applied to find the 

 equations of equilibrium for the simpler case of a single homogeneous 

 assemblage there considered. Thus we have an incompressible elastic 



* This result was given for an equilateral tetrahedronal assemblage in 67 of 

 " Molecular Constitution of Matter/' 'Math, and Phys. Papers,' vol. iii, pp. 425 

 426. 



