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Proceedings of Royal Society of Edinburgh. [sess. 
every atom separately of the whole assemblage. The total amount 
of work required to separate all the atoms to infinite distances 
from one another is This (not subject to any limitation 
such as that stated for the former procedure) is rigorously true for 
any assemblage whatever of any number of atoms, small or large. 
It is, in fact, the well-known theorem of potential energy in the 
dynamics of a system of mutually attracting or repelling particles ; 
and from it we easily demonstrate the item ±nW 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. 
Tor 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 there- 
fore have = N?tf, and therefore the whole work required to 
separate all the atoms to infinite distances is — 
JN w (1). 
§ 4. Let <£(D) he the work required to increase the distance 
between two atoms from D to oo ; and let /(D) he the attraction 
between them at distance D. We have 
/(D)=-A^(D) (2). 
For either assemblage I. or assemblage II. we have 
w = </>(D) + </>(D') + <£(D") + etc (3); 
where D, D', D", etc., denote the distances from any one atom of all 
neighbours, including the farthest in the assemblage, 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- 
