119 



is, therefore, always a centrically-symmetrical arrangement, and as 

 the existence of this symmetry is equivalent to a symmetry-property 

 of the second order, it follows from this, that a space-lattice can 

 never differ from its mirror-image. Its symmetry belongs in all 

 cases to that of the symmetry-groups of the second order, and more 

 especially to those amongst them, which are characterised by the 

 possession of a symmetry-centre. Of course this fact will at once 

 restrict appreciably the number of eventually possible symmetrical 

 arrangements of this kind. 



A second universal property of space-lattices is, that an eventual 

 symmetry-axis of it must always be parallel to, or coincident with a point- 

 row of the space-lattice; and moreover each symmetry-axis must be 

 always perpendicular to a net- 

 plane of the space-lattice too. 



The truth of both these facts 

 can easily be deduced from some 

 simple geometrical reasonings. 



Finally it will be clear that, if 

 a space-lattice has a symmetry- 

 axis of the period y, = , not 



passing through a point of the 

 system, it must have simul- 

 taneously an infinite number of Fig. 105. 

 parallel symmetry-axes of the 

 same period passing through every point of the space-lattice. 



The truth of this can be demonstrated as follows. Let S (fig. 105) 

 be the point of intersection of a symmetry-axis A(a) with the plane 

 of drawing; this plane, according to what is said above, is certainly 

 a net-plane of the space-lattice, and therefore, P l may represent a 

 point situated in it nearest to 5. If we turn the space-lattice 



r\ 



round A (a) through tx, = , the point P, comes into P 2 , and P 2 



must, therefore, be also a point of the system. If this is now shifted 

 along PiP 2 , until P 2 coincides with P lf the point of intersection 5 

 will have reached S', while the point P 1 will have returned to its 

 original position. Both successive operations are evidently together 

 equivalent to a rotation about an axis passing through P lt which 



brings S in 5', the period /3 of this axis also being = . It is 

 demonstrated in this way, that there are really at all points P 



