116 



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 frofn 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 always 



be 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 a, = - , not 



n 



passing through a point of the 

 system, it must have simul- 

 Fj g I0 , taneously an infinite number of 



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. foj) 

 be the point of intersection of a symmetry-axis A(x) 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 of it situated in it nearest to 5. 



If we turn the space-lattice round A(u) through a = , 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 PjP 2 , until P 2 coincides 

 with P 1( the point of intersection 5 will have reached S' ' , while the 

 point Pj_ will have returned to its original position. 



Both successive operations are together evidently equivalent 

 to a rotation about an axis passing through P lt which brings 5 



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



n 



