382 Professor G. D. Liveing [May 15, 



so long as their parallelism is maintained ; but the orientation will 

 aifect very mucli the symmetry of the crystal. 



If we suppose the spheroids to be oblate, and arrange them as in 

 Fif^ure 1, with their axes perpendicular to the plane of that figure, and 

 place the next layer in those triangular openings which are white in the 

 figure, and complete the pyramid, the magnitude of the three angles at 

 the apex of the pyramid will depend on the relative flatness of the 

 spheroids. In case the length of the axis of the spheroids is half their 

 greatest diameter, these three angles will be right angles, and the 

 whole heap of molecules will have a cubic symmetry, and in the faces 

 of the cubes the concentration will be a maximum, and therefore the 

 surface energy a minimum, and the easiest cleavage will be cubic. If 

 the concentration in the cubic faces be 1 * 0000, that in the octahedral 

 faces will be • 5774, and that in the dodecahedral • 7071. We have 

 here the case of crystals like rock salt and galena. Suppose, however, 

 we start with the arrangement of Figure 3, and keep the axes per- 

 pendicular to the plane of that figure ; and suppose, further, that the 

 biggest diameter of the spheroids is greater than the length of the 

 axis in the ratio of the diagonal to the side of a square, we shall 

 again get a heap with a cubic symmetry ; but in this case the maxi- 

 mum concentration will be in the faces of the dodecahedron, and 

 we have the case of blende in which the easiest cleavage is dodeca- 

 hedral. 



In order to see what the symmetry will be in other cases, we may 

 look at the problem from another point of view. Suppose a cube made 

 up of spherical molecules to be subject to a uniform stress perpendicular 

 to one face of the cube, so that all the spheres are strained, either 

 by extension or compression, into spheroids, we should get that 

 dia^^onal of the octahedron which was parallel to the stress either 

 lengthened or shortened, but the symmetry about that diagonal would 

 remain as before. We should get a crystal of the pyramidal system. 

 If the spheroids were prolate and sufficiently elongated, the easiest 

 cleavage would be perpendicular to the axis as in potassium ferro- 

 cyanide and apophyllite. If the spheroids were oblate the funda- 

 mental octahedron would be more obtuse, and if obtuse enough the 

 easiest cleavage would be in faces parallel to the axis of symmetry. 



Attain, if the stress, instead of being perpendicular to one face of the 

 cube, were parallel to a diagonal of the cube, the cube would become 

 a rhombohedron, and the spheres would become spheroids with their 

 axes parallel to the axis of the rhombohedron. If the spheroids were 

 prolate the rhombohedron would be acute, and the easiest cleavage 

 perpendicular to the axis as we find it in beryl and many other 

 crystals. If the original cube were formed of spheroids with their 

 axes half the length of their greatest diameters, and the stress parallel 

 to the axes were such as to alter the length of the axes only a little, 

 we should get crystals with a rhombohedral cleavage like calcite. 

 The crystals like beryl almost always exhibit hexagonal forms, six- 

 sided prisms and pyramids. To explain this I would observe that if 



