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Proceedings of Royal Society of Edinburgh. [sess. 
guide line through A. The distances from A of all the atoms 
in this net are, according to fig. 1, 
+ + . . . ( 6 ). 
All the other atoms of the equilateral assemblage to which A 
does not belong lie in nets at successive distances k, 2k, 3k, 
etc., beyond the two nets we have already considered on 
the two sides of our “ middle plane ” ; the atoms of each 
net placed of course according to the cyclical law described in 
(e) of § 6. 
§11. Working out for the double assemblage II. for A’s nearest 
neighbours according to § 10, we find four nearest neighbours 
at equal distances = *61 3A ; twelve next-nearests at equal 
distances A ; and twelve next-next-nearests at equal distances 
x / 3 g 1 A = 1T73 A. These suffice for § 12 below. It is easy and 
tedious, and not at present useful, to work out for D 4 , D 5 , D 6 , etc. 
§ 12. Using now §§ 9, 11 in (3) of § 4 we find, — 
for assemblage I., 
ic = 120(A) + 60(1 ’414A) + 180(1‘732A) + 60(2A) + . .! 
for assemblage II., f ' 
io = 40(*613A) + 120(A) + 120(1 -173A) + 
These formulas prepare us for working out in detail the practical 
dynamics of each assemblage, guided by the following statements 
taken from §§ 18, 16 of M. C. M. 
§ 13. Every infinite homogeneous assemblage of Boscovich atoms 
is in equilibrium. So, therefore, is every finite homogeneous 
assemblage, provided that extraneous forces be applied to all 
within influential distance of the frontier, equal to the forces 
which a homogeneous continuation of the assemblage through 
influential distance beyond the frontier would exert on them. 
The investigation of these extraneous forces for any given homo- 
geneous assemblage of single atoms — or groups of atoms as ex- 
plained above (§ 1) — constitutes the Boscovich equilibrium- theory 
of elastic solids. 
It is wonderful how much towards explaining the crystallo- 
graphy and elasticity of solids, and the thermo-elastic properties 
of solids, liquids, and gases, we find; without assuming, in the 
Boscovichian law of force, more than one traijsition from 
