WILLIAM HALLOWES MILLER. 4G5 



substance ; * and Miller called these relative magnitudes the parameters 

 of the crystals, while he called the whole numbers, vi, n, and p, the 

 indices of the respective planes. But, instead of writing the propor- 

 tion which expresses the law of crystallography as above, he gave to 

 it a slightly different form, thus, — 



A: B :C=-i-^ a:-^b : jc, 



and used in his system for the indices of a plane the values h : k : I, 

 which are also in the ratio of whole numbers, and usually of simpler 

 whole numbers than m : n : j). This seems a small difference ; for hkl 

 ill 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 formuhe which express the relations between the parts of a crys- 

 tal. From the last proportion we derive at once 

 I 



1 a 1 b 1 c 



1 ' A 'k ' D~T ' IJ 



which is the form in which Miller stated his fundamental law. 



If P represents the "pole" of a face whose "indices" are hhl, 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, T, Z represent the 

 points where the axes of the crystal meet the same spherical surface, f 

 then it is evident that XT, X Z, and T Z Ave the arcs of great circles, 

 which measure the inclination of the axes to each other, and that P X, 

 P T, and P Z are arcs of other great circles, which measure the inclina- 

 tion of the plane (h k I) 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 



-^ cos PX = ^ cos PT= J cos PZ 



and by means of this theorem we are able to reduce a great many 

 problems of crystallography to the solution of spherical triangles. 



* For example, the native crystals of suli^luir have a : b : c = l : 2.340 : L233, 

 Crystals of gypsum liave a : b : c=l : 0.413 : 0.691, 



Crystals of tin stone have a : b : c=l : 1 : 0.6724, 



And crystals of common salt have a : b : c = I : I : 1. 



t The origin of the axes is always taken as tlie centre of the sphere of 

 projection. 



