A : B : C = m a : n b : p c,
in which m, n, and p are simple whole numbers. The irrational values a, b, and c are fundamental magnitudes for every crystalline substance;[G] and Miller called these relative magnitudes the parameters of the crystals, while he called the whole numbers, m, n, and p, the indices of the respective planes. But, instead of writing the proportion which expresses the law of crystallography as above, he gave to it a slightly different form, thus:
| A : B : C = | 1 h | a : | 1 k | b : | 1 l | c, |
and used in his system for the indices of a plane the values h : k : l, which are also in the ratio of whole numbers, and usually of simpler whole numbers than m : n : p. This seems a small difference; for h k l in the last proportion are obviously the reciprocals of m n p in the first; but the difference, small as it is, causes a wonderful simplification of the formulæ which express the relations between the parts of a crystal. From the last proportion we derive at once
| 1 h | · | a A | = | 1 k | · | b B | = | 1 l | · | c C | , |
which is the form in which Miller stated his fundamental law.
If P represents the "pole" of a face whose "indices" are h k l, that is, represents the point where the radius drawn normal to the face meets the surface of the sphere circumscribed around the crystal (the sphere of projection, as it is called), and if X, Y, Z represent the points where the axes of the crystal meet the same spherical surface,[H] then it is evident that X Y, X Z, and Y Z are the arcs of great circles, which measure the inclination of the axes to each other, and that P X, P Y, and P Z are arcs of other great circles, which measure the inclination of the plane (h k l) on planes normal to the respective axes; and, also, that these several arcs form the sides of spherical triangles thus drawn on the sphere of projection. Now, it is very easily shown that
| a h | cos P X | = | b k | cos P Y | = | c l | cos P Z, |
and by means of this theorem we are able to reduce a great many problems of crystallography to the solution of spherical triangles.
Another very large class of problems in crystallography is based on the relation of faces in a zone; that is, of faces which are all parallel to one line called the zone axis, and whose mutual intersections, therefore, are all parallel to each other. If, now, h k l and p q r are the indices of any two planes of a zone (not parallel to each other), any other plane in the same zone must fulfill the condition expressed by the simple equation