156 



PRINCIPLES OF CRYSTALLOGRAPHY, 



symbols li Ti l, and the other half by efg. The forms of this system are 

 generally pyramids of twenty-four faces, two orders of twelve-faced pyra- 

 mids, whose faces are perpendicular to the principal section, prisms of 

 twelve faces, two orders of six-faced i^risms, and the base. 

 7. Tesseral (isometric) system. — Nine planes of symmetry, three 



of which, A A' A", (Fig. 26,) 

 are similar and at right an- 

 gles to each other; the others 

 tautozonal in pairs, with an 

 A intercalated between each 

 two, B . . . . B^, at an angle 

 of 45° to them. We select 

 the three which are perpen- 

 dicular to each other for the 

 plane of the axes, and deter- 

 mine the length of the axes 

 by the face 111, which lies 

 in an intermediate zone; we 

 t '"'» thus have — 



a'= & = c; e = >y = C = 90o 

 The five elements are determined. 



The most general form, hJcl, consists of forty-eight faces, whose dis- 

 tribution is shown in Fig. 26. 



In the previous development only the most general form, JiTcl, has 

 been considered ; it is, however, very easy by specializing the symbols, 

 as by an equation of two indices, for instance, or by conditions which can be 

 conceived in the projection, to represent all the forms of a system by 

 the number and signs of the faces. 



We wish, for example, the symbol of the faces of the six-sided, the 

 twelve-faced pyramid of the hexagonal system. Their symbols result 

 from the relation of the zones. On the other hand, a simple inspection 

 of the planes of symmetry of this system shows that a face occurring 

 in the zone [(111) (2 1 1) j has on the upper side five similar faces. Thus 

 it results that the symbol of the opposite rhombohedron, similar to the 

 primitive rhombohedron, is (122), according to the formula, (p. 22.) 

 Partial forms have not been included in the above representation, any 

 more than the researches on the symmetry of lines and planes, which 

 will be given in another place. 



SECTION III. 



optical relations of crystals. 

 § 1. — Double Eefraction and Absorptiotst. 



It is known that in media of equal density throughout, also in uncrys- 

 tallized media, a ray of light moves in every direction with the preserva- 

 tion of its condition of vibration; that, further, its velocity of propagation 



