250 PRINCIPLES OF CRYSTALLOGRAPHY. 



or the general symbol li li I ; and lias [1 1 IJ for its zone-index, so — 



h-\-Ji~l = 0ov2h = l 

 which condition is satisfied by (112). 



Thus the collective forms of this combination are determined. 



There certainly may be cases presented where the existing zones do 

 not suffice to determine all the faces of a combination ; these cases are, 

 however, rare, and occur in very few instances. 



Instead of the above selection of a face, (111), determining the collect- 

 ive relations of the axes, two domes in two i)inacoid zones could very well 

 be used, as 1 1 0, in which a : h, and 101, by which a : c,is determined. 



In the simpler and often recurring faces, as we have seen above, even 

 the very simple calculation of the symbols from two zone-symbols, by 

 crosswise multiplication, is superfluous, because at least the conditions 

 for it, in a zone, can be at once expressed in the general symbol of the 

 face, so that by substitution in the equation — 



Jix + ley -\-ls = 

 the indices hlcl are fully determined. 



SECTION II. 



symmetry of the systems of crystallization. 



§ 1.— Derivation of the Sy^stem from the Law of K ational Indices. 



The rationality of the indices is, for the possibility of a face of a 

 crystal, as we have said above, not only a necessary but a sufficient 

 condition. It is, therefore, a possibility of every face whose indices are 

 rational numbers. A collection of faces, therefore, which is to obey the 

 law of rational indices must also answer to all the consequences which 

 in mathematics follow from this law. 



The carrying-out of this deduction, which can here only be announced, 

 leads us to the different elements of symmetry, and especially to the 

 consideration of planes of symmetry. 



A plane of symmetry has the peculiarity that its physical relations 

 are equal on both sides of it. 



The identity of the physical peculiarities of two faces or lines is also 

 jr,;^j3,^ determined by the similarity of their position with 



respect to the plane of symmetry, and this condition 

 is really fulfilled by two planes when they are tau- 

 togonal with the plane of symmetry, and are so sit- 

 uated with regard to both sides that they form like 

 anzles with them, (Fig. 13a, where the angle P:Q = aO 

 and pi : Q = /3o are equal to each other.) Two lines, 

 A and o B, (Fig. 13&,) satisfy the condition if they, 

 with respect to the plane of symmetry P, contain a 

 similar angle; and if a plane, E, at right angles to the 

 plane of symmetry can be passed through, then arc A C = arc B. 



The derivation of the crystalline system is as follows : Let two possi- 

 ble faces of a crystal be taken which are symmetrical with respect to a 



