30 ME. H. J. BEOOKE ON THE GEOMETEICAL ISOMOEPHISM OF CETSTAXS. 
The angles quoted in the accompanying Tables, and throughout this paper, are those 
between normals to the several faces, although, to avoid the frequent repetition of the 
words “ between normals,” they are expressed as angles between faces. The angle between 
any two faces, as given by most other authors, and the angle between normals to those faces, 
are supplements to each other, so that, if either is given, the other can be readily found. 
A zone in crystallography is a series of faces on any cr^’stal which intersect each other 
in parallel lines, or would do so if sufficiently enlarged. 
It has been found that the secondary faces of crystals are not produced as it were at 
random, but that they cut the primary edges in definite proportions measured along those 
edges in particular directions ; and it is by means of the differences of those proportions, 
and of the directions in which they are measm'ed, that secondarj- faces can be denoted 
and distinguished from each other. This system of notation was first proposed by the 
Rev. Dr. Whewell, in a paper read before the Royal Society and pubhshed in the Philo- 
sophical Transactions for 1825, and has been employed by Professor Millee of Cam- 
bridge in a work on Crystallography pubhshed in 1839, and in the new edition of the 
Treatise on Mineralogy, by the late William Phillips. 
According to this system, the faces of a primary form, as they 
are usually shown in engraved figures, are expressed by the sjrni- 
bols 10 0, 010, 001, placed in the relative positions shoum in 
fig. 1 ; and the faces respectively parallel to these are denoted by 
Too, OTO, OOT. 
These numbers, taken separately, and regarding 0 or zero as a 
number, are termed indices, and those with a line over them are 
termed negative or minus. 
It is seen in the skeleton (fig. 2) of the primary form, that the 
edge between the faces 0 01 and 010, and the three edges parallel 
to these, have an m at one end and m at the other ; that the edge 
between 0 01 and 10 0 and the three parallel edges have n and n 
on each edge ; and that the edge between 10 0 and 010, and the 
three parallel edges, have s and i on each edge. 
By means of these indices an appropriate symbol can be assigned 
to every possible face of every crystal, an advantage not afforded by 
any other system of notation. 
Suppose a secondary face to occur, as in fig. 3, on the solid angle 
surrounded by the letters mns, and to be such as would cut away 
^ of the edge m, ^ of n, and ^ of s, measured along each respective 
edge from the point where m, n, and s meet. This face is denoted 
by the symbol 3 2 4; the separate indices being the reciprocals of 
the proportions supposed to be cut away from the respective edges 
by this particular face. 
If a face truncates that solid angle of fig. 3 which is in fig. 2 
Fig. 1. 
Fig. 2. 
Fig. 3. 
