of a Crystal according to Boscovich. 429 



shall suppose. And the corner C of the equal and dichirally 

 similar tetrahedron OA'B'C is one of the points in the nearest 

 parallel plane below. All the points in the plane through C 

 are corners of equal tetrahedrons chirally similar to OABC, 

 and standing on the horizontal triangles oriented as BOA. 

 All the points C in the nearest plane below are corners of 

 tetrahedrons chirally similar to OA'B'C placed downwards on 

 the triangles oriented as B'OA'. The volume of the tetra- 

 hedron OABC is ^ of the volume of the parallelepiped, of 

 which OA, OB, 00 are conterminous edges. Hence the sum 

 of the volumes of all the upward tetrahedrons having their 

 bases in one plane is -jt of the volume of the space between 

 large areas of these planes : and, therefore, the sum of all the 

 chirally similar tetrahedrons, such as OABO, is ^ 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 

 corners, 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 correspond- 

 ing positions relatively to the primary assemblage. 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 tetra- 

 hedron, 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 infinitesmal change of shape possible to 

 it ; so that we have, in fact, the skeleton of an incompressible 

 and inextensible solid*. Let now any forces whatever, sub- 



* 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. 



