115 



Bet. ') lints it </, >rf 2 > d 3 , the meshes of the net-plane (d t , dj 



will have a smaller an a than those of the net-plane (rf,, d 3 ) or (d lt d t ), 

 and mure particularly will these areas be inversely proportional to 

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

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



nsity 5 can be expressed by -, in which f is the surface of the 



uallt'logram-shaped mesh of the net-plane. From what has been 

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

 roportional to the distance between two of them in Die same parallel 

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

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

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

 their surface-density. 



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

 chosen, the distances of the successive points along them being 

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

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

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

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

 itercepts on the three axes OX, OY, and OZ segments, the length 

 >f which are always of the form: md lt nd z , and rd s respectively, m, n, 

 ind r being such integer numbers too. 



The significance of these considerations for the special problem 

 >f crystal-structure is evident, because the fact just mentioned 

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

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

 >revious Chapter). 



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

 ice-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 

 ime 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 and 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. 



