ME. H. J. BEOOKE ON THE GEOMETEICAL ISOMOEPHISM OF CETSTALS. 31 
surrounded by w, n, s, and intersects m at ^ of its length measured from the point 
where m, n, s meet, and if it theoretically removes ^ of the edge n, and a of s, measured 
from the same point, its symbol would be 3 2 4. 
An analogous face on the solid angle m, n, s would have for its symbol 3 2 4, and 
another similar face on the solid angle m, n, s would have the symbol 3 2 4; and if we 
suppose a similar face to truncate the solid angle m, n, s, its symbol would be 3 2 4. 
A face that would cut away equal proportions of each of the edges m, n, s, would have 
for its symbol 111, and the face parallel to it 1 1 1 ; and the symbol of a face parallel to 
any given face is denoted by simply changing the positive indices of the given face to 
negative, and the negative indices to positive. 
The faces which replace the edges of crystals being always parallel to those edges, do 
not anywhere intersect them, and in consequence the index relating to the replaced edge 
is always 0. 
Thus in fig. 4 the edge m is entfrely removed, while ^ of n and of s are theoretically 
cut away by the truncating face, whence its symbol becomes (0 2 4). The symbol of the 
face in fig. 5 is similarly 3 0 4, and of that in fig. 6 is 3 2 0. 
Thus recollecting that the symbols of primary faces have two zeros, and of faces on 
the edges one, the position of any face denoted by any symbol may be readily perceived 
and easily remembered. 
It may be noticed, that the order in which the indices stand in each of the preceding 
symbols is that of the letters m, n, s ; and in the symbol of any face whatever, the first 
index always denotes the reciprocal of the proportion supposed to be cut away from one 
of the edges m ; the second index, the reciprocal of the proportion removed from one of 
the edges n ; and the thfrd, the reciprocal of the proportion removed from one of the 
edges s. 
Faces may be expressed by their symbols, as face 1 0 0 or 1 1 1. 
An angle between two faces may be expressed by means of their symbols, as the angle 
between 10 0 and 111, or simply the angle 10 0, 111. 
A zone may be expressed by the symbols of any two faces in it, as (1 0 0, 1 1 1). 
There are in each of the systems, except the cubic, angles between particular faces by 
which the crystals belonging to the different minerals contained respectively in each are 
distinguished from each other. These will be termed elementary angles. 
F 2 
