70 Lord Kelvin. On the [June 15, 



Thus we have the remarkable result that, relatively to the principal 

 cube, the diagonal rigidity is half the facial rigidity when each point 

 experiences force only from its twelve nearest neighbours. This pro- 

 position was announced without proof in (28) of " Molecular Con- 

 stitution of Matter."* 



24. Suppose now the points in the middles of the faces of the 

 cubes which in the equilateral assemblage are O's twelve equi- 

 distant nearest neighbours to be removed, and the assemblage to 

 consist of points in simplest cubic order ; that is to say, of Bosco- 

 vichian points at the points of intersection of three sets of equidistant 

 parallel planes dividing space into cubes. Fig. 2 shows O ; and, at 

 X, Y, Z, three of the six equidistant nearest neighbours which it has 

 in the simple cubic arrangement. Keeping A, with the same signifi- 

 cation in respect to fig. 2 as before, we have now for the coordinates 

 of O's six nearest neighbours : . 



(Xv/2, 0, 0), (0, \V2, 0), (0, 0, \v/2), 

 (-Xv/2, 0, 0), (0, -\</2, 0), (0, 0, -AV2). 



Hence, and denoting by uri the value of nr for this case, we find, by 

 18 (2*), 



A. = Nw! and B = w = ........... (31). 



The explanation of n = (facial rigidity zero) is obvious when we 

 consider that a cube having for its edges twelve equal straight bars, 

 with their ends jointed by threes at the eight corners, affords no 

 resistance to change of the right angles of its faces to acute and 

 obtuse angles. 



25. Replacing now the Boscovich points in the middles of the 

 faces of the cubes, from which we supposed them temporarily 

 annulled in 24, and putting the results of 23 and 24 together, 

 we find for our equilateral homogeneous assemblage its elasticity 

 moduluses as follows : 



(32), 



B = n 



where, as we see by 16 (20) above, 



<ar = XF(X) X 2 F'X 



(33), 



i 2X 2 F'(X v / 2) ' 



F(r) being now taken to denote repulsion between any two of the 

 points at any distance r, which, with 0(r) defined as in 10, is the 



* Math, and Phys. Papers,' vol. iii, p. 403. 



