140 Lord Kelvin on 



dynamics of a system of mutually attracting or repelling 

 particles ; and from it we easily demonstrate the item J?? W 

 in the former procedure. 



§ 3. In the present communication we shall consider only 

 atoms of identical quality, and only two kinds of assemblage. 



I. A homogeneous assemblage of N single atoms, in which 

 the twelve nearest neighbours of each atom are equidistant 

 from it. This, for brevity, I call an equilateral assemblage. 

 It is fully described in M.C. M., §§ 46, 50 . . . 57. 



II. Two simple homogeneous assemblages of £N" single 

 atoms, placed together so that one atom of each assemblage 

 is at the centre of a quartet of nearest neighbours of the 

 others. 



For assemblage II., as well as for assemblage I., w is the 

 same for all the atoms, except the negligible number of those 

 within influential distance of the boundary. Neglecting 

 these, we therefore have Sw=Nu?, and therefore the whole 

 work required to separate all the atoms to infinite distances is — 



PTw (1). 



§ 4. Let cf>(D) be the work required to increase the distance 

 between two atoms from D to x ; and let /(D) be the 

 attraction between them at distance D. We have 



/(D) = -^*(D) (2). 



For either assemblage I. or assemblage II. we have 



w=:£(D)+0(DO+<MD'') + etc. ... (3); 



where D, D', D", etc., denote the distances from any one 

 atom of all neighbours, including the farthest in the assem- 

 blage, which exercise any force upon it. 



§ 5. To find as many as we desire of these distances for 

 assemblage I. look at figs. 1 and 2. Fig. 1 shows an atom A, 

 and neighbours in one plane in circles of nearest, next-nearest, 

 next-next-nearest, etc. Fig. 2 shows an equilateral triangle 

 of three nearest neighbours, and concentric circles of neigh- 

 bours in the same plane round it. The circles corresponding 

 to i\ and r s of § 7 below, are not drawn in fig. 2. In all 

 that follows the side of each of the equilateral triangles is 

 denoted by X. 



§ 6. Ail the neighbours in assemblage I. are found by aid 

 of the diagrams as follows : — 



(a) The atoms of the net shown in fig. 1. The plane of 

 this net we shall call our "middle plane." Let lines be 



