Mr. W. Barlow on Crystal Symmetry. 31 



This completes the list of types of identical repetition about 

 a centre presented when there is a plurality of axes. A 

 similar method to that employed on page 22 leads to the dis- 

 covery of a homogeneous structure corresponding to each of 

 them. The types are, as stated above, the six shown by 

 figs. VI.-X1., in which no mirror-image similarity is re- 

 presented. 



Adding the five cases previously described, there are in 

 all eleven types when mirror-image repetition of directions 

 similarly related to the structure is excluded; viz.: one anorthic 

 and ten which exhibit different directions identically related 

 to the structure. 



Mirror-Image* Similarity of Parts. 



As already stated f the similarity of corresponding parts 

 may amount to identity, or it may be mirror-image resem- 

 blance. To complete the survey it is therefore necessary to 

 turn to the cases in which a similarity of parts exists which 

 is a mirror-image one, the corresponding parts present, 

 whether points similarly placed or directions similarly situated, 

 being divisible into two sets so connected that the relation of 

 the entire system to the individual points or directions of the 

 one set is the mirror-image of its relation to the individual 

 points or directions of the other set. 



As the aspect of such a system when viewed from one 

 standpoint within it is the mirror-image of its aspect as 

 viewed from some other standpoint, it is evident that corre- 

 sponding to every set of linear directions and parts which 

 bear an identical relation to the structure there is an equally 

 numerous set of directions and parts whose relation to the 

 structure is similar but not identical to that of the members 

 of the first-named set, and which, like the latter, are identical 

 among themselves. And it follows also that the figure of the 

 entire system of repetition is, in a case of this kind, irre- 

 spective of orientation, identical with the figure of its mirror- 

 image. 



Let some complete set of identically related directions be 

 located in a system possessing the property described, care 

 being taken not to select those occupying any specially 

 symmetrical position. It will then be possible also to indicate 

 a corresponding equally-numerous set of directions whose 

 relation to the system is similar, but not identical, to that of 

 the directions of the first set; and it follows from the fore- 

 going that the arrangement of the one set of directions is 



* See note 2, p. G. f See p. 6. 



