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set. l ) Thus, if d^ ]> d 2 > d 3 , the meshes of the net-plane (d z , d 3 ) 

 will have a smaller area than those of the net-plane (d lt d^ or (d lt d 2 ), 

 and more particularly will these areas be inversely proportional to 

 the distances d lt d z , and d 3 respectively. When we define the density 

 of a net-plane as the number of meshes per unit of surface, this 



density can be expressed by , in which s is the surface of the 



parallelogram-shaped mesh of the net-plane. From what has been 

 said, it is obvious that the surface-density of net-planes is directly 

 proportional to the distance between two of them in the same parallel 

 set. The further such parallel net-planes are distant from each other, 

 the greater will their surface-density be, and conversely: the distance 

 between two parallel net-planes of a space-lattice is directly proportional 

 to their surface-density. 



In the second place, if three axes of reference OX, OY, and OZ 

 be chosen, the distances of the successive points along them being 

 d lf d 2 , and d 3 respectively, then the coordinates of every point P 

 of the space-lattice will always be pd lt qd 2 , and sd 3 , in which p, q, 

 and s are integer numbers. It may be easily conceived from this, how 

 this fact involves that every net-plane passing through the point P 

 intercepts on the three axes OX, OY, and OZ segments, the lenghts 

 of which are always of the form: md lt nd 2 , and rd 3 respectively, m, n, 

 and r being also such integer numbers. 



The significance of these considerations for the special problem 

 of crystal-structure is evident, because the fact just mentioned 

 is a direct geometrical expression of Hauy's law in crystallography, 

 concerning the rational indices of possible crystal-facets (see the 

 previous Chapter). 



5. We will now examine the symmetry-properties of such 

 space-lattices more closely, and in the first place draw attention 

 to some general facts in connection with this symmetry. 



Because every point of the endless space-lattice has wholly the 

 same function as all other points, it must be at once clear that 

 every point of a space-lattice is a symmetry -centre of it. A space-lattice 



!) It can easily be demonstrated, that primary cells, i. e. such cells of the 

 space-lattice which do not contain any other points within their parallelepiped 

 volume besides those at the corners of it, have always the same minimum volume. 



A simple geometrical reasoning will convince us of this, and a similar 

 thesis is valid for the primary meshes of a net-plane, i. e. for each mesh which 

 contains no more points within its parallelogrammatical area. 



