18 Mr. W. Barlow on Crystal Symmetry. 



divisible geometrically into identical finite portions which are 

 similarly -orientated *. 



Proof. If the structure has no axis its coincidence-move- 

 ments are all translations. If, on the other hand, axes are 

 present, its possession of translations at right angles to these 

 axes and in other directions may be established as follows : — 



Among the axes present, some are, by proposition 10, 

 identical and parallel. Let A, A' be two such axes. Then, 

 if a certain screw-movement, or rotation, proper to A, carries 

 a point P to P l5 a similar movement will characterize A' and 

 will carry the same point P to a third point P 2 ; and the points 

 P l5 P 2 thus located from P will lie in a plane perpendicular 

 to the axes and will be identically related to the structure ; 

 and. further, the aspect of the structure viewed from P 2 will, 

 since both coincidence-movements have the same rotation 

 component, present the same orientation as the identical 

 aspect viewed from P 2 ; in other words, the structure will 

 be capable of a coincidence translation from P 2 to P 2 . 



The employment of other pairs of the parallel axes will lead 

 in the same way to the discovery of various other coincidence 

 translations perpendicular to these axes. 



If the system possesses in addition axes having other 

 directions, these also involve the existence of various transla- 

 tions perpendicular to them. If, on the other hand, it does 

 not, identical points not lying in the same plane perpendicular 

 to the axes must be capable of being brought to coincidence 

 either by screw-spiral movements about the axes which are 

 present, or by translations ; and as the former have translation 

 components along the axes, there must, in all cases, be other 

 translations present besides those perpendicular to the one 

 set of axes. 



Therefore all homogeneous structures, both those wholly 

 destitute of axes, and those possessing them, are capable of 

 coincidence translations in various directions not lying in a 

 single plane direction. The fundamental definition on which 

 this inquiry is based requires these translations to be all finite f . 



* Unless the symmetry is anorthic these are not the space-units of 

 page 7. They will each, when as small as possible, consist of as many 

 contiguous space-units of a certain pattern as are found differently orien- 

 tated. They need not have the same degree of symmetry as that of the 

 structure, although in the majority of cases the partitioning for obtaining 

 them can be so carried out as to give them this symmetry. 



f See p. 5. There are cases of homogeneous repetition in which the 

 translations are not all finite. The structure of columnar basalt, if 

 regarded as quite uniform, is a case in point — the translations in one 

 direction become infinitely small and, although such a structure possesses 

 an infinite group of coincidence-movements it is not available, not being- 

 divisible into finite physical units ; it does not follow Hatty's, law. 



