242 



PEINCIPLES OF CRYSTALLOGRAPHY. 



If the planes of the axes are parallel faces, X o Y, Y o Z, Z o X, as the 

 faces ABO and H K L, which may be real or possible faces of a crys- 

 tal, experience shows that the indices h Tc I of every possible face of this 

 crystal are to each other as rational numbers. 



This law, which is the first fundamental law of crystallography, is 

 called the law of rational indices ; it is of the greatest importance, and 

 allows of the derivation of the greater part of the other laws of crystal- 

 lography. 



If the indices li k I of any face of a ^crystal are rational, it is always 

 possible to represent them by three positive or negative whole numbers, 

 because the direction of a plane remains unchanged when its three 

 indices are multiplied by the same number. 



Experience shows further that the indices of the most frequently 

 occurring faces are almost always the vSimplest whole numbers and 1, 

 rarely 2, so that the calculation with them will always be very simple. 



§ 2. — Law op Zones. 



The consideration of the zones occurring in a crystal is of the greatest 

 importance for the determination of a combhiatiou. 



Two planes which are not parallel always cut each other, when duly 

 extended, in a straight line ; all planes, therefore, whose lines of section 

 are parallel to the same straight line, belong to a zone, and are called 



Fog.5 



tautozonal faces : the straight line to 



which their lines of section are parallel 

 is called the axis of the zone. (Fig. 5.) 



Because the axis of a zone is parallel 

 to all the faces of that zone, a plane, P, 

 perpendicular to the axis of the zone, will 

 also be perpendicular to all the faces of 

 that zone ; and when a perpendicular to 

 every zone-fnce is erected, all of these 

 normals will be parallel to this face P. 

 This important characteristic of tau- 

 tozonal faces, that their normals all 

 lie in a i)lane perpendicular to the zone- 

 axis, we shall make use of in the discussion of spherical projection. 

 After the direction of the zone-axis is determined by the section of two 

 planes which are not parallel, it must be possible, from the known ele- 

 ments of these planes, to calculate for the indices such values as will be 

 characteristic for the axes of the zone produced by these planes. Let 

 P {hhl) and Q [pqr) be the two planes, and let their indices be written 

 twice, one over the other, and multiplied crosswise, beginning with the 

 second upper index k — 



Ji Ic I h Tc I 



XXX 

 p qr p q r 



Tir — Iq; Ip — hr ; h q 

 u V 



Jcp 



w 



