170 



with respect to a centre or a molecular row of the space-lattice, 

 the direction of it being other than the axis of the constitutive 

 particle itself, in the sense of Bravais' theory, (fluorspar; 

 fig, jjj). This is valid for holohedral, as well as for merohedral 

 forms; in the first case however Mallard has shown that the sym- 

 metry-axis of the new compound crystal 

 as a whole, can also be an axis of approxi- 

 mate symmetry of the space-lattice under 

 consideration, in the sense in which we 

 have defined it in the preceding chapter, 

 and that a special tendency to the sym- 

 metrical arrangement of the separate in- 

 dividuals round this axis of approximate 

 symmetry is observable in these cases l ). 

 In connection with this it must be remar- 

 ked that such twins can also have a plane 

 of symmetry, and in the case of holohedral 

 crystals, a plane of approximate sym- 

 metry. This is easily understood if one considers that, if an axis of 

 even period and an inversion-centre be the symmetry-elements of 

 twins, the existence of a symmetry with respect to a plane perpendi- 

 cular to that axis also is of course involved as a logical consequence. 

 Resuming we may say that experience has shown there is a 

 remarkable tendency in merohedral crystals to twin-formation, 

 in such a way that as preferential twinning-elements there appear 

 such planes or such axes of even periods, as in the holohedral class 

 of the same crystal-system have the function of true symmetry- 

 planes or of true symmetry-axes (pyrite). The twins appear therefore 

 as an approximation to the holohedral symmetry of the system 

 to which the crystal belongs, and according to Haidinger, they 

 may be given the name of completion-twins (calamine, quartz). And, 

 secondly, experience teaches that planes and axes of approximate 

 symmetry may have the function of twinning-elements too ; in this 



Fig. 133- 

 Fluorspar. (Twin.) 



1) E. Mallard. Ann. des Mines. 20. (1876); Bull, de la Soc. Miner. 8. (1885); 

 Revue Scientifique. (1887); Cf. also: A. Bravais, Etudes Cristallographiques, 

 Paris, (1866), p. 248. 



It may be understood, that twins are single individuals. If there be spoken 

 here and in the preceding paragraphs, of two or more individuals which form 

 "compound" twins, then this is simply a mode of speech, used for the purpose 

 to help imagination. 



