22 Mr. W. Barlow on Crystal Symmetry. 



classes of centred symmetry of identically related parts may 

 be noted : — 



A Te7t£) (n °} *ow„ i„ (PL I.) fig. MGadolinfig.W). 

 A single digonal axis ,., ,, II. ( „ 41) 



A single trigonal axis „ ,, III. ( ,, 53). 



A single tetragonal axis „ „ IV. ( „ 35).. 



A single hexagonal axis „ „ V. ( ,, 50). 



All these types of symmetry are found represented in 

 homogeneous structures, since, corresponding to each of them, 

 a type o£ structure exists which repeats through space the 

 order of symmetry of the centred type"*. To obtain such a 

 type of structure proceed as follows : — 



Construct a space-network of points (Raumgitter) whose 

 symmetry is or includes that of one of the types of centred 

 symmetry. Then place at the points the centres of a number 

 •of identical bodies whose symmetry is of the type in question, 

 taking care that these bodies are similarly and appropriately 

 •orientated. Thus — to give a simple instance — let a space 

 network be formed whose points lie at all the angles of a 

 ■number of identical regular triangular prisms fitted together 

 symmetrically to fill space. Then place sameway-orientated, 

 •with their centres at the points, and their axes perpendicular 

 to the triangular layers of the network, a number of identical 

 bodies having a trigonal axis .and no other, e. g. coins of the 

 Isle of Man with the threefold Manx emblem on them. The 

 system obtained by using the latter presents, if the obverses 

 of the coin be neglected, a case of trigonal symmetry with a 

 single axial direction. 



The combination of each of the above five kinds of repeti- 

 tion of identically related directions with mirror-image 

 repetition will be considered later. 



Cases where there is a Plurality of Directions of 

 Coincidence- Axes. 



Proposition 18. — Where axes in vnore than one direction 

 are present, axes in at least two directions have the same 

 rotation-angle. 



For if some two of the mutually-inclined axes are of 

 different rotation-angles one of these angles must be less than 

 180°, and when the rotation proper to such an axis is made at 

 least two directions for the other description of axis will be 

 discovered. 



Proposition 19. — Where there is a plurality of directions 



* This kind of structure is presented byBravais's assemblages. 



