12 Mr. W. Barlow on Crystal Symmetry. 



lines located in this way in the centred figure must have, 

 when the latter is appropriately orientated, precisely the 

 directions of those located in the homogeneous structure, and 

 will therefore display the same arrangement of like directions. 



In the case of anorthic symmetry, the rotation components 

 of the coincidence movements are all zero and so no two 

 directions are identically alike, all the coincidence move- 

 ments being mere translations. 



As the experimental methods at the disposal of the crystal- 

 lographer are not at present refined enough to discover in 

 a direct manner any discontinuity of substance in crystals, 

 variation of property with direction, not variation from point 

 to point, is all that can be discriminated in these bodies, and 

 therefore the only parts of the coincidence movements 

 referred, to which are immediately traceable in actual crystals 

 are the rotation components, which, as just stated, determine 

 the nature of the symmetrical repetition of identically related 

 directions. Consequently, for the purpose of ordinary 

 crystal classification, these rotation components, not the 

 translation components, are of significance, and the tracing 

 of merely directional symmetry is all that is necessary. 

 While it is an arduous and extended work to go further than 

 this, and to ascertain all the varieties of type which can exist 

 of the groups of coincidence movements of homogeneous 

 structures, it is a much simpler and shorter task to investigate 

 merely the different rotation components of these movements 

 and their relative arrangement. The former, more extended 

 inquiry, which has had thus far rather a theoretical than a 

 practical utility, has been carried out very fully by various 

 investigators *. The latter, more limited one, taken in con- 

 junction with the principle of mirror-image repetition f, leads 

 to the derivation of the 32 classes of crystal symmetry from 

 first principles, and it alone is the object of the present 

 investigation. 



To arrive at the 32 classes a method is here adopted which 

 combines some of the arguments employed by Sohncke with 

 some of those used by Gadolin and others. 



The following ten propositions lead, up to the solution of 

 the question what varieties of axes are possible in homo- 

 geneous structures ; several of these propositions are taken 

 with some slight modifications from Sohncke J. 



* See Entwickelung einer Theorie der Krystallstructur von Dr. Leonhard 

 Sohncke, Leipzig, 1879; Schonflies' Krystallsysteme u. Krystallstructur ; 

 11 Theorie der Krystallstructur/' von E. von Fedorow, in Zeitschrift fur 

 Krystallographie, xxiv. p. 209 ; and also see Zeit. f. Kryst. xxiii. p. 1, and 

 xxv. p. 86. t See below p. 31. 



\ Entwickelung einer Theorie der Krystallstruktur, p. 37. 



