OK THE STRUCTURE OF CRYSTALS. 080 



classes, was arrived at by William Barlow- ^ by a somewhat different 

 reasoning. 



The sixty-five i^oint-systems of Sohncke are of two sorts : some of them 

 are identical with their own mirror-images and some are not. Further, there 

 are some very simple homogeneous structures which are not fuJJy repre- 

 sented among Sohncke's systems. An example will be given directly. 



Barlow designates the identical symmetrical repetition of parts through- 

 out space, as investigated by Jordan and Sohncke; by the title ' homo- 

 geneous structure,' and his definition of such a structure is very similar to 

 that given by Fedorow of a regular system of figures. He shows that the 

 point-systems obtained by taking all the homologous points in such a 

 structure are Sohncke's point-systems, and that every such structure is 

 capable of the coincidence-movements of some one of Sohncke's sixty-five 

 systems and of no other. 



Suppose, for example, a number of equal cubes to be stacked together 

 in the most regular manner, and let any geometrical point be taken within 

 one of the cubes. There are within this cube twenty-threeother points, at 

 the same distance from its centre as the first, which have identically - the 

 same relation to the whole stack, so that the latter presents the same 

 aspect when viewed from any one of the twenty-four points. 



These twenty -four points constitute a Sohnckian 24-punktner, and when 

 corresponding points are taken in all the cubes Sohncke's system 59 is 

 obtained. 



By a method of developing structures of higher symmetry from those 

 of lower symmetry, Barlow obtains all Sohncke's sixty-five sets of coin- 

 cidence-movements, and points out that corresponding to each of these 

 sixty-five systems is a class of homogeneous structure which is not identical 

 loith its own tnirror-iriiage. He then remarks that the additional 

 property of identity with mirror image can be displayed by homogeneous 

 structures in a definite number of different ways, and that this enables 

 us to distinguish other types of symmetry besides the sixty -five types 

 established without this property. 



For example, let a line be drawn from each point of one of the 

 24-point groups above described through the nearest cube-centre, and 

 prolonged to an equal distance on the opposite side. The twenty-four 

 points thus obtained, together with all similar points, constitute a second 

 Sohncke-system which is the mirror image of the first ; from each of them 

 the aspect of the structure as a whole is the mirror-image of its aspect 

 from any one of the first set. The two together represent fully the true 

 symmetry of the stack of cubes, which is thus shown to possess a higher 

 symmetry than the simple Sohncke-system derived from it. 



If, now, space is to be filled with similar unsymmetrical cells, instead 

 of cubic cells, one such cell must enclose each of the first system of pointr-, 

 and another which is its mirror-image must similarly enclose each of the 

 second system of points. In the original paper the diagrams of symme- 

 trical partitioning which are introduced make this conclusion easier to 

 follow. The cells clearly correspond to the Fundmnentalbereiclie of 



' ' Ueber die geometrischen Eigenschaftenhottiogener starrer Structuren nnd ihre 

 Anwendung auf Kryst&WQ,' Zi-its. Kryst. Min., 1834, vol. sxiii. pp. 1_63 ; and 1895, 

 XXV. p. 86. 



- Points having a mirror-image relation to the point selected are not here taken 

 Into consideration. 



