Mr. W. Barlow on Crystal Symmetry. 7 



presented by which will be that of the different orientations of 

 an identical aspect of the structure as viewed successively from 

 some set of homologous standpoints icithin it. 



PjROOF. For let the positions o£ some such set of identi- 

 cally repeated geometrical points be noted, the set selected 

 being one which is as numerous as any other such set, and 

 let every one of these points be similarly enclosed by a cell 

 of arbitrary, but identical shape, the relative orientations and 

 positions of the cells being such that the aspect of every cell 

 and of the ultimate structure viewed from the enclosed point 

 is the same ; and provided that the cells are small enough to 

 avoid interpenetration. 



Next let all these cells expand uniformly and similarly in 

 every direction*, but only till neighbouring cells are encount- 

 ered, two moving cell-boundaries which approach one another 

 coming instantaneously to a standstill directly they touch. 

 As the conditions about all the corresponding points are pre- 

 cisely the same, the result evidently is to completely fill space 

 with enlarged cells which, like the original cells, are all 

 identical, and the mass displays the property that the aspect 

 of the enclosing cell and of the ultimate structure is the 

 same from each of the corresponding points, although the 

 identical aspects, like the cells, may be variously orientated. 

 The partitioning achieved is therefore that required. 



Fie. 4. 



* As this is only a geometrical operation, the fact of space being- 

 occupied with the matter of the homogeneous structure is, of course, no 

 barrier to it. 



