536 
Thus, a horizontal layer (suppose) may be arranged in square , or 
in triangular order; and successive layers maybe simply superposed 
or inserted into interstices in the preceding ones. 
By this it would appear, at first sight, that arrangements of four 
different densities are producible. Such, however, is not the case. 
The very obvious results which I intend now to give, must have 
occurred long ago to others, but I have found no such record ; and 
their apparent novelty to scientific friends to whom I have men- 
tioned them, must be my excuse for occupying the time of the So- 
ciety with so trifling a matter. 
The density of a mass, or the number of marbles per cubic mile 
(say) is easily seen to be proportional inversely to the product of 
the distances of the centres of contiguous ones, taken parallel to any 
three rectangular lines. 
Hence 
(1.) When layers in square order are superposed, a being 
the radius of a molecule — 
Densityoc 2 ^^<t« oc h 
(2.) Layers in triangular order superposed — 
Density * 2a x 2a x * 4^. 
(3.) Layers in square order, interstices occupied by mole- 
cules of next layer — 
Density cc 
1 
2a x 2a x s]2a 4:.sJ2a 3 
(4.) Layers in triangular order, interstices as before, - 
1 1 
Density oc 
<x 
2a x x 
2 s l2a 4V2a 3 
"73 
1 /2 
These densities are (1.) : (2.) : (3.) : (4.) : : — : \/ - : 1:1; 
V & o 
or, '707 : '816 : 1 : 1, and may be sought for in various forms of 
the same body. 
Hence, the density in arrangements (3) and (4) is the same. It is 
worthy of note, that the simplest crystalline form which is derivable 
from (3), is the octahedron of the first system, and that from (4), 
is its hemihedral form, the regular tetrahedron. 
