34 CRYSTALLOGRAPHY. 
2¢:2:1,4¢:2:1, 5¢:3:1, and so on; or, giving it a general form, 
mes: 1. 
In the lettering of the planes on figures of dimetric crystals, the first 
number (as in the isometric and all the other systems) is the coefficient 
of the vertical axis, and the other is the ratio of the other two, and 
when this ratio is a unit it is omitted. 
The expressions and the lettering for the planes are then as follows: 
Expressions, Lettering. 
F LONgen esd 4-0 
or square prismS..:........ ab vigees ded sable 
For eight-sided prisms......... 40 de 1-2 
For octahedrons...........- | : one ; 1 ; : se 
For double eight-sided pyramids, me: n:1 m-n 
The symbols are written without a hyphen on the figures of crystals. 
On figure 14, the plane 7-7 is that particular 2-n in which n = 2, or 7-2. 
In fig. 21 the planes of the double eight-sided pyramid, m-n, bave 
m — 1 and 2 — 2 (the expression being 1 : 2: 1), and hence it is lettered 
1-2. In fig. 8 and in fig. 22 it is the one in which m=3 andn=3 
{the expression being 3: 3: 1), and hence the lettering 3-3. 
The length of the vertical axis ¢ may be calculated as follows, pro- 
vided the crystal affords the required angles: 
Suppose, in the form fig. 18, the inclination of O on plane 1-7 to have 
been found to be 139°, or of id on the same plane, 140° (one follows 
from the other, since the sum of the two, as has been explained, is 
necessarily 270°). Subtracting 90°, we have 40° for the inclination of 
the plane on the vertical axis ¢, or 50° for the same on the lateral axis 
a, or the basal section. [In the right-angled triangle, OBC, the angle 
OBC equals 40°. If OC be taken as a = 1, then BC will 
24. be the length of the vertical axis c; and its value may be 
y obtained by the equation cot 40° = = BC, or tan 50° = BC. 
On fig. 18 there is a second octahedral plane, lettered 
4-1, and it might be asked, Why take one plane rather 
than the other for this calculation? The determination 
on this point is more or less arbitrary. It is usual to 
assume that plane as the unit plane in one or the other 
series of octahedrons (fig. 16 or fig. 17) which is of most 
common occurrence, or that which will give the simplest 
symbols to the crystalline forms of a species; or that 
which will make the vertical axis nearest to unity; or 
0 © that which corresponds to a cleavage direction. 
The value of the vertical axis having been thus deter- 
n ined from 1-7, the same may be determined in like manner for 4-7 in 
the same figure (fig. 18). The result would be a value just half that of 
BC. Or if there were a plane 2-7, the value obtained would be twice 
BC, or BD in fig. 24; the angle ODC + 90° would equal the inclina- 
tien of O on 2-7. So for other planes in the same vertical zone, as 3-2, 
4-7, or any plane m-?. 
If there were present several planes of the series 72-7, and their incli- 
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