120 



such axes parallel to the one supposed in 5, and that all have 

 the same period. 



With respect to the possible combinations of symmetry-elements 

 in such space-lattices, we can refer here to the contents of the prece- 

 ding chapters // to IV \ the general rules stated there are valid 

 also here. The only question yet to be considered is: what can be 

 the periods of the axes of symmetry in such space-lattices? 



Let P (fig. iotf) be a point of 

 the system, Let us suppose that a 

 symmetry-axis A (tx) of the period 



r\ 



passes through P, and that it is 



perpendicular to the plane of the 

 figure. According to the above, 

 it is, therefore, at the same 

 time a net-plane of the space- 

 lattice. The point situated nearest 

 to P in this net-plane may be N l . 

 When we perform now the characteristic rotations round A through 

 angles a, 2a, 3a, etc., the point A^ reaches successively the corres- 

 ponding points N 2 , N 3 , NI, etc., of the net-plane. But because of the 

 parallelogram-shaped meshes of this net-plane, a point Q must also be 

 found in the net-plane in such a way that Q, N lf N 2 , and N 3 together 

 form a primary mesh of it. Moreover, the coordinates of all these 

 points in the net-plane must be in rational proportion to each other. 

 Now we have supposed that N-^ was nearest to P; the absolute 

 distance PQ may therefore only be greater, or in the extremest case 



be equal to PN lt etc. Now j-. is evidently ==1 4 sin 2 f-j; and 



if we calculate the values of this expression for n = 3, 4, 5, 6, etc., 

 we obtain the following result: l ) 



106. 



l ) Of course, the number 2 is certainly valid here, as can immediately be 

 seen from a simple figure. 



