114 



OP 3 may be taken as axes of reference OX, OY, and OZ; then the 

 parallelopipedon having d lr d z , and d 3 as its edges, is the absolutely 

 determined, parallelepiped "unit-cell" of the infinitely extended 

 space-lattice, and evidently no other point of the latter is situated 

 within this parallelepiped cell any longer. The whole space-lattice 

 might also be imagined to be built up by three sets of an infinite 

 number of net-planes, all parallel to and equidistant from the three 

 pairs of opposite limiting faces of the parallelepiped cell ; and in the 

 same way an infinite number of sets of parallel equidistant net- 

 planes can be distinguished in the space-lattice, all made up by 

 points placed at the corners of parallelogram-shaped meshes, while 

 no other points are situated within the boundaries of these parallel- 

 lograms. 



The essence of a space-lattice is that it is a homogeneous and perio- 

 dical structure of points, in which each point therefore is situated 

 relatively to its neighbours in exactly the same way as every other 

 point. The parallelepiped unit-cell represents the "geometrical 

 period" of the space-lattice, and this period, although extremely 

 small, is always a finite one. The orientation of every net-plane 

 therein, is determined by the space-lattice alone; and to every 

 net-plane there corresponds a set of an infinite number of congruent 

 net-planes, all parallel to and equidistant from the first. Moreover 

 the assemblage may possess special symmetry-properties by which 

 the individual shape of the unit-cells and their marshalling are 

 determined; in such a case the points will have a perfectly regular 

 geometrical arrangement in which the various symmetry-elements 

 will be associated according to the general rules of the doctrine 

 of symmetry, as deduced in the preceding chapters. 



4. Before dealing with these symmetry-properties of space- 

 lattices, it is of interest to consider some of their general proper- 

 ties in detail. 



In the first place it is clear that the meshes of the various 

 net-plane.s of a space-lattice are of different sizes, but constant 

 for every net-plane of a certain situation. The parallellograms in 

 the net-planes parallel to the coordination-planes, as determined 

 above, evidently possess the three smallest areas which can occur 

 at all in the space-lattice under consideration. Because the unit -cell 

 of smallest volume has a constant volume, this surface of the 

 meshes will be smaller in the same rate, as the distance between 

 the equidistant net-planes of the same set is greater than in another 



