oor> 



REPORT — 1901. 



mentioned above, which possess respectively three, four, six, and seven such 

 pairs of faces. 



The following example illustrates the principles described above, and 

 is of special interest as representing a tj'pe consistent with the symmetry 

 of the mineral dioptase which first led to an extension of Sohncke's 

 theory. This mineral possesses an axis of trigonal combined with a 

 centre of symmetry, its faces always occurring in sets of six, which are 

 all alike, and consist of thi-ee pairs of parallel faces. (The symmetry 

 may equally well be described as due to the operation of an hexagonal 

 axis of alternating symmeti'y.) 



Fig. 10 represents a system of stereohedra arranged in accordance 

 with this symmetry: the stereohedra are of two sorts (R and L), one sort 

 being the mirror-image of the other ; the structure is symmorphous. 

 A series of points similarly situated, one within each stereohedron 

 R, would constitute a Sohncke-system : a ' double system ' of points is 

 obtained by adding a series similarly situated, one within each stereo- 

 hedron L. The figure also shows the manner in which the stereohedra 



Fig. 10.' 



can be grouped in sets of six to form parallelohedra, which in this case 

 are rhombohedra. Consequently a rhombohedral partitioning of space is 

 consistent with the given type of symmetry. 



From the principles laid down in the memoirs mentioned above, 

 Fedorow is able to deduce all the possible types of symmetry which 

 characterise either homogeneous systems of figures, or homogeneous 

 systems of points, or the homogeneous partitioning of space. 



They are 230 in number, and are identical with those established 

 independently, as stated above, by Schonfiies. 



Fedorow's theory of crj'stal structure, which is based upon his 

 parallelohedra, will be considered later. 



Barlow. 



The result attained by Fedorow and Schonfiies, that homogeneous sti'uc- 

 tures, if classified by their symmetry, can be distinguished not only into 

 thirty-two classes but into 230 kinds which belong to these thirty-two 



' The stereohedra are shown in the figure slightly drawn apart to make the 

 arrangement clearer, but iu £30*; they fill space without interstices, 



