24. CRYSTALLOGRAPHY. 
tained by adding 90°. If m =1, then the ratio is i: 1, as in ACB, 
and each angle equals 45°, giving 135° for the inclination on either 
adjoining cubic face. 
Again, if the angles of inclination have been obtained by measure- 
ment, the value of n in any case may be found by reversing the above 
calculation ; subtracting 90° from the angle, then the tangent of this 
angle, or the cotangent of its supplement, will equal 2, the tangents 
varying directly with the value of 7. 
In the case of planes of the m: 1: 1 series (including1:1:1, 2:1: 
1, ete.), the tangents of the angle between a cubic face in the same 
zme and these planes, less 90°, varies with the value of m. In the 
case of the plane 1 (or 1: 1 : 1), the angle between it and the cubic face 
is 125° 16’. Subtracting 9)°, we have 35° 16. Draw a right-angled 
triangle, OBC, with 35° 16’ as its vertex angle. BC has 
jy the value of ic, or the semi-axis of the cube. Make 
DC=2BC. Then, while the angle OBC has the vaiue 
of the inclination on the cubic face less 90° for the plane 
1:1:1, ODC has the same for the plane 2:1:1. Now, 
making OC the radius, and taking it as unity, BC is the 
tangent of BOC, or cot OBC. So DC = 2BC is the tan- 
gent of DOC, er cot ODC. By lengthening the side CD 
(= 2BC or 2c) it may be made equal to LLC =3ce, its 
value in the case of the plane3:1:1; or to4BC = 4e, 
its value in the case of the plane 4:1:1; or mbC=me 
0 © for any plane in the series m:1: 1; and since in all 
there will be the same relation between the vertical and 
the tangent of the angle at the base (or the cotangent of the angle at 
the vertex), it follows that the tangent varies with the value of m. 
Hence, knowing the value of the angle in the case of the form 1 
(1: 1:1), the others are easily calculated from it. 
BC being a unit, the actual value of OC is 4 4/2, or 4/z. it being half the 
diagonal of a square, the sides of which are 1, and from this value the 
angle 35° 16’ might be obtained for the angle OBC. 
‘The above law (that for a plane of the m7: 1 : 1 series, the tangent of 
its inclination on a cubic face lying in the same zone, less 90°, varies 
with the value of m, and that it may be calculated for any plane 
m:1:1from this inclination in the form 1:1:1), holds also for 
planes in the series m:2:1, or m: 3:1, or any m:n:1. That is, 
given the inclination of O on 1:7: 1, its tangent doubled will be that 
of 2: 2:1, or trebled, that of 8: 2:1, and so on; or halved, it will be 
that of the plane $: 2:1, which expression is essentially the same as 
ley 2n G2: 
These examples show some of the simpler methods of applying ma- 
thematics in calculations under the isometric system. The values of 
the axes are not required in them, because a =b=c=1. 
iss 
3. Hemihedral Crystals.—The forms of crystals described 
above are called holohedral forms, from the Greek for ali and 
fice, the number of planes being all that full symmetry re- 
quires. The cube has eight similar solid angles—similar, that 
is, in the enclosing planes and plane angles. Consequently the 
law of full symmetry requires that all should have the same 
