
OIMEThIC, OR TETRAGONAL SYSTEM. 35 
tatious to the basal plane O were known, then, after subtracting from 
the values 90°, the cotangents of the angles obtained, or the tangents 
of their complements, will equal m in each case; that is, the tangents 
(or cotangents) will vary directly with the value of m. The logarithm 
of the tangent for the plane 1-2, added to the logarithm of 2, will 
equal the logarithm of the tangent for the plane 2-2, and so on. 
The law of the tungents for this vertical zone m-? holds for the planes 
of all possible vertical zones.in the dimetric system. Further, if the 
square prism were laid on its side so that one of the lateral planes be- 
came the base, and if zones of planes are present on it that are vertical 
with reference to this assumed base, the law of the tangents still holds, 
with only this difference to be noted, that then one of the lateral axes 
is the vertical. It holds also for the trimetric system, no matter which 
of the diametral planes is taken for the base, since all the axial inter- 
sections are rectangular. It holds for the monoclinic system for the 
zone of planes that lies between the axes ¢ and } and that between the 
axes @ and 06, since these axes meet at right angles, but not for that 
between ¢ and a, the angle of intersection here being oblique. It holds 
for all vertical zones in the herugonal system, since the basal plane in 
this system is at right angles to the vertical axis. But it is of no use 
in the ¢riclinic system, in which all the axial intersections are oblique. 
The value of the vertical axis c may be calculated also from the incli- 
nation of O on 1, or of Jon 1. See fig. 2, and compare it with fig. 17. 
If the angle J on 1 equals 140°, then, after subtracting 90°, we have 50° 
for the basal angie in the triangle OCB, fig. 24; or for half the inter- 
facial angle over a basal edge—edge Z—in fig. 17. The value of ¢ 
may then be calculated by means of the formula 
c= tan} Zy4, 
by substituting 50° for $Z and working the equation. 
For any octahedron in the series m, the formula is 
me =—tan4Z v4 
Z being the angle over the basal edge of that octahedron. If m = 2, 
then c= 4(tan4Z v4). Further, m = (tan 4Z v3) +. 
The interfacial angle over the terminal edge of any octahedron im 
may be obtained, if the value of ¢ is known, by the formulas 
me = cote cos e = cot 4X 
X being the desired angle (fig. 17). The same for any octahedron m-¢ 
may be calculated from the formulas 
me = cote cose=cosiY v2 
Y being the desired angle (fig. 16). 
For other methods of calculation reference may be made to the *‘ Text 
Book of Mineralogy,” or to some other work treating of mathematical 
crystallography. 
g. Hemibcdral Forms.— Among the few hemihedral forms 
wnder the dimetric system there is a tetrahedron, called a sphen- 
