222 Prof. Lnveing, On Solution and Crystallization. [May 21, 
which cuts the three axes distances p, g, 7 respectively from the 
origin, where p, g and r are whole numbers, will be a surface of 
maximum concentration of molecules, but the concentration will 
be less as p,q and 7 are greater. Hence forms which are bounded 
by these planes, which follow the law of indices of erystals, will 
be forms of minimum surface energy and therefore of equilibrium. 
The tendency in general will be for substances with such a struc- 
ture as is here supposed to take the form of cubes, since the cube 
will have the greatest concentration of molecules per unit of 
surface. But the total surface energy will depend on the total 
surface as well as on the energy per unit of surface, and for a 
given volume the surface will be diminished if the edges and 
angles of the cube are truncated by faces of the dodecahedron 
and octahedron, or by more complicated forms. 
When a solid is broken two new surfaces are formed each with 
its own surface energy, and the solid must be more easily frac- 
tured when the new surfaces have the minimum energy. Hence 
substances with the structure supposed must break most easily 
in directions parallel to the sides of the cube, dodecahedron and 
octahedron: and these are the cleavages observed in this system. 
If we suppose the molecules placed at the centres of the faces of 
the cubes, instead of at the angles, the arrangement will still be 
isotropic, but the octahedron will be bounded by the surfaces of 
greatest condensation and the cube will come next to it. It is 
probable that substances which cleave most readily into cubes, 
such as rock salt and galena, have the former structure, while 
those which have the octahedral cleavage may have the latter 
arrangement of their molecules. 
For the pyramidal and prismatic systems we may suppose 
space divided not into cubes but into rectangular parallelopipeds 
with edges equal severally to the axes of the crystals, and mole- 
cules placed as before. For the rhombohedral system we may 
suppose space divided into rhombohedra, or in crystals of the 
hexagonal type into right prisms with triangular bases, and for 
the other systems into parallelopipeds with edges parallel and 
equal to the axes. In each case if the molecules be disposed at 
points of intersection of three dividing planes we shall have such 
an arrangement as satisfies the optical conditions, and planes 
which follow the law of indices are surfaces of maximum con- 
densation. Calculations show that whenever a crystal has an 
easily obtained cleavage the direction of cleavage corresponds to 
the surface of greatest condensation, and that the most common 
forms of crystals correspond in general to forms of minimum 
surface energy. 
The surface tension of a plane surface will have no resultant 
out of that plane, but where two plane surfaces meet in an edge, 
