32 



Mr. W. Barlow on Crystal Symmetry. 



the mirror-image of that of the other set. This is true of 

 enantiomorphously corresponding directions, whether occur- 

 ring in homogeneous structures or in their spheres of 

 reference*. As an example of the latter : — the set of similar 

 normals, whose situation is shown in (PI. I.) fig. XIV., may be 

 regarded as made up of the set indicated in (PL I.) fig. III., 

 and a set which is the mirror-image o£ the latter. 



Now whether a system of symmetrical repetition about a 

 centre, which is not a centre of inversion, is identical with its 

 own mirror-image or not, it is always possible to find a 

 system of a higher order of symmetry which presents all the 

 symmetry of the given system and whose centre is a centre 

 of inversion; and the presence of such a centre produces an 

 additional symmetrical, but not identical duplication of all 

 the repetitions of the system taken. For the invert of this 

 system with respect to its centre has the same axes, and the 

 elements of symmetry of the two systems thus related can 

 therefore exist together in a more complex type of repetition 

 having the same coincidence-axes as the two simpler systems, 

 and with the centre of inversion added. If the original 

 system is not identical with its mirror-image, the additional 

 repetitions found in the corresponding system which has a 

 centre of inversion will be all identical and will be the mirror- 

 image of all those of the simpler system. If, on the other 

 hand, the simpler system is already identical with its own 

 mirror-image, though not possessed of a centre of inversion, 

 the additional repetitions, like those of the simple system, will 

 be separable into two sets, one of which is the mirror-image 

 of the other. As an example of the former case the type 

 shown in fig. XIV., which has a centre of inversion, duplicates 

 the type shown in fig. III.; and, as an example of the latter 

 case, the type shown in fig. XXII. duplicates that of fig. 

 XXXII., the latter being identical with its own mirror-image. 



But although some centred types which contain a centre 

 of inversion may, in this way, be derived from types that are 

 identical with their own mirror-image, they, as well as the rest 

 of the types possessing such centres, can also always be re- 

 garded as derived by the duplication of a system of repetitions 

 which has not this property. For, in every case where a 

 centre of inversion is present, just half the similarly related 

 directions or parts are identically and not merely similarly 

 placed with reference to the entire structure; and therefore 

 these, taken alone, possess all the coincidence-movements of 

 the structure and exhibit one of the types of symmetry not 



* See p. 11. 



