THE VARIATION OF ANOLES ORSERVED IN CRYSTALS. 
-Hi 
no longer be expressed by simple rational indices, but only by very bigli numbers. 
Cubes ot duor, for example, will often yield no reflections corresponding\o • the true 
faces of a cube, and, therefore, do not possess angles of 90°; but each apparent cube 
face really consists of a very flat pyramid whose adjacent planes are inclined to each 
other at the angle 2° 31f. These faces would intersect the axes of the cube in the 
ratio 1 : 32, and belong to the form (32.1.0). 
Such vicinal faces are encountered upon crystals of most substances ; sometimes 
they are produced by the solvent action of tlie mother-liquor; but they may certainly 
also appear during the growth of the crystal wlien it is not being redissolved. Tlie 
vicinal faces themselves are sometimes as perfectly plane and smooth as any on the 
crystals. 
In the case of fluor, the presence of the group of four such planes' in place of the 
cube face shows that they are not merely due to the distortion of a cube face, Imt 
belong to a different form. But even when a face with liigh indices occurs u|)on a 
crystal as a single isolated plane, it is generally regarded as a vicinal face and either 
as due to the etcliing action of a solvent, or as replacing a face with simpler indices 
owing to tlie operation of some unknown cause. 
Whetlier tlie vicinal faces really obey the law of rational indices at all is not 
crnfain. ^Planes to which it is necessary to attribute high indices are usually called 
“ vicinal,” and are regarded as something different from ordinary faces ; many careful 
measurements of individual vicinal planes have been made in order to determine their 
indices, whereas it is the custom to eliminate variations in the angle between what 
appear to be faces of simple forms. Sometimes, however, tliese variations are so large, 
even in Bie case of what appear to be c[uite simple forms, that tliey can scarcely'lie 
ignored in this manner. 
In most of the crystalline systems the theoretical angles of tlie crystal are not 
known, but have to be calculated from some of the observed angles, which may be 
themselves liable to these variations, so that it is difficult to say whether the variations 
are really irregular. But m the cubic system, by virtue of its symmetry, the angles 
are known absolutely, and it is possible to compare the measured angles with Bie 
theoretical values. Crystals belonging to the cubic system appear, however, to be 
liable to^ the same variations of angle with those of other systems, and also Exhibit 
vicinal faces. 
Now the whole value of the law of rational indices, which is the foundation-stone 
of crysffallograi)hy, rests upon tlie permanence of angle. The faces of an octahedron 
belonging to the cubic system are inclined to each other at 70° 31' 44”; and, similarly, 
the angle of the form (112), of which any face cuts two of the axes of the octahedron 
at their extremities and bisects tlie third, is 38° 56' 33". If the octahedron and 
icositetrahedron faces do not make these angles, respectively, then the law of rational 
indices is only a hist ajiproximation, and there is some disturbing influence which has 
yet to be investigated. 
