PEINCIPLES OF CRYSTALLOGRAPHY. 



241 



If another face, A B 0, is given, with the parameters o A, o B, o G, 

 which we may call a, h, c, then — 



A = a ; o B = h ; o C = c 



and the face H K L is determined when the relations — 



A 



n = ij^ = 



a 

 6~E 



oB 



Jc = '^ = 



oC 





c 

 oh 



oH^oH'oK oK'"~oL 

 are known ; so a third face, H' K' L', is determined by its relations or 

 indices h' Jc' l', in which — 



« - 7,/ b , y _ c 

 oK" 



li = 



H" 



V 



oh' 



We see, also, that if three planes, X o Y, Y o Z, Z o X, are given, 



77^.4, 



whose three lines of section 

 represent the axes o X, o Y, 

 o Z ; farther, a fourth face, 

 A B 0, whose section of these 

 axes is the measure of the 

 length of the axes, any face 

 in their direction is perfectly 

 determined when its indices, 

 i. e., the relation between the 

 parameters of A B C and its 

 own, are given. 



The values a b c and the x 

 plane of the axes are constant for one and the same crystal. 



Eespecting the indices h, Jc, I, certain important cases are to be dis- 

 tinguished : 



I. All three of the indices may be different from (o h, Ar, I,) ^ o. This 

 is the general case, and represents octahedral or pyramidal faces. 



II. One index, I, for instance, equals zero, 1 = o ; the face h, Ic, o, is 

 evidently parallel to the axis o Z, and we have — 





qC c 



h~ oh 



Because o C = c is constant, this fraction can only be equal to o if o L 

 is infinitely great ; but if the face H K o cuts the axis o Z at an infinite 

 distance, it is parallel to it. Thus, if fc = o, we have h o I, and if h = o, 

 we have o Ic I oi the axis of Y, parallel faces with respect to X. These 

 kinds of faces are called dodecohedral, prismatic, or dome-faces. 



III. Two indices = o1c = 1 = o. .. .100; l = 1i=o = 010; fi = Jc = o 



01, the face 10 has first the index 1c = o, and is for that reason 



parallel to the above axis of Y, and also to the axis of Z, because I = o. 

 This face contains, therefore, both the axes of Y and Z. It is with them 

 parallel to the axis-plane X o Z. We call such faces pinacoids ; they 

 are those by means of whose section-line the position of the axes is deter- 

 mined. 



16 s 



