ON tse structure of crystals. 317 



(1, 2, 3, 01' 7i) be present, the component assemblage formed by each 

 kind, taken by itself, being homogeneously arranged, and all the difi'erent 

 kinds possessing identical systems of axes and having the same set of 

 translations common to them.' 



Sohncke's aim is, as has been said, to produce the requisite varieties 

 of symmetry by arranging regular or spherical particles homogeneously; 

 This he succeeds in doing by his enlarged method, and is now able to 

 cover the cases of hemimorphism.^ Instead, however, of merely stipulating 

 that the component point-systems shall have the same translations common 

 to them, and possess identical systems of axes, he ought to have stipulated 

 that they shall have all their coincidence-movements in common.'^ 



For all the coincidence-movements which characterise the combined 

 system as a whole must obviously be obeyed by every particle within it, 

 and it is only these movements which really belong to the component- 

 systems as found in the structure. In other words, if there are other 

 coincidence-movements in addition to these, which a set of points would 

 have if taken alone, such movements must for the combined systein be 

 regarded as non-existent, and onhj those j^oints of such a set will have 

 identical positions in the entire system ivhich can be brought to coincidence 

 by the surviving movements, i.e., by those which characterise the structure 

 as a whole. After making this distinction it will usually be possible to 

 detect two or more different kinds of points forming two or more different 

 subsidiary point-systems, which must be counted separately, as many 

 systems being discriminated as there are varieties of position of the 

 points. When this is done the various different point-systems present 

 ■will have all their coincidence-movements in common, these movements 

 being those characteristic of the combined system as a whole. 



Reference to an example may make this clearer to those who are 

 familiar with Sohncke's treatise. Let two point-systems (a and b) be 

 taken, each of which, when regarded apart from the other, presents the 

 same instance of type No. 2 of Sohncke, and which have their systems of 

 axes and their translations identical ; let them be combined in such a 

 way that they are sameway- orientated and have the two sets of points 

 lying in the same planes, but with the axes distinct. See fig. 7, in which, 

 to distinguish the two systems, one (b) is represented in dotted lines. 

 Either system consists of a series of equidistant parallel planes, each 

 beset with particles in the same way ; and the diagram is one such plane ; 

 the points in the succeeding planes lie vertically below those in the 

 diagram. Then the combination thus formed must be regarded as con- 

 sisting of four separate point-systems, not of two only, for the positions in 

 the composite structure occupied by the points are ot four different kinds. 

 Each of the four sets is destitute of axes ; the composite system has 

 merely the symmetry which it would have had if constructed of four 

 distinct point- systems, each possessing the translations common to the 

 two initial systems, and consisting of points lying in the same planes. 



In Sohncke's work rigid geometrical results are closely interwoven 



' Zcits. Kryst. Mm., 1888, vol. xiv. p. 433. Comp. nh., 1892, vol. xx. p. 450. 



- For further applications of his method see 'Zwei Tbeorien der Krj'stallstruclur,' 

 Zclts. Kryst. Min., 1892, vol. sx. p. 455. 



^ Sohncke was disposed at first to make this stipulation, but did not perceive its 

 necessity; he afterwards definitely adopted the less precise one to which objection 

 is here taken. Comp. Zcits. KryU. i)/ut., 1888, vol. xiv. p. 441. and 1892, vol. sx. 

 p. 45G. 



