123 



indejK'Mdfnt ol tli<-ir ;u ( idental qualities, was solved by 

 l for the cases in which only syimnetry-propertfes <-i the 

 order were considered; afterwards the complete solution, in< lu- 

 ilso the symmetry-properties of the second order, was given 

 jy Von Fedorow 2 ) and bij Schoenflies 3 ), while similar stu- 

 lies on the principle of homogeneity were published by Barlow 4 ) 

 id others 6 ). Of course, as soon as tridimensional arrangement- 

 considered, which have also symmetry-properties of the second 

 order, the necessity arises of adopting the possibility in such systems 

 of two kinds of "motifs" which are enantiomorphous with respect 

 to each other. For by the operations of the second order characteristic 

 )r the tridimensional pattern, each motif is converted into its 

 lirror-image ; and as soon as the motif itself is deprived of all qua- 

 ities, and therefore of all specific symmetry, its mirror-image must 



in general non-superposable with itself. 



Therefore homogeneous systems in space, possessing also symmetry- 

 >roperties of the second order, must be built up by two enantio- 

 morphously related kinds of repeats, and only such patterns as are 

 themselves different from their mirror-images, i. e. which possess 

 only symmetry-properties of the first order, are in general formed 

 by the regular arrangement of one and the same kind of pattern-units. 

 The regular structures, as deduced by Sohncke, are completely 

 determined by rotations and translations', the latter and their com- 

 binations with certain motions about axes of the first order, which 



1) ~L. Sohncke, Entwickelung einer Theorie der Krystallstruktur, Leipzig, 

 (1879); Wied. Ann. der Physik. 16. 489. (1882); Zeits. f. Kryst. 13. 214. 

 (1888); 14. 417, 426. (1888); Pogg. Ann. d. Phys. 137. 177. (1869). 



C. E. Von Fedorow, Symmetric der regelmassigen Systeme von Figuren 

 (1890); Zeits. f. Kryst. 20. 25. (1892); 24. 209. (1895); 25. 113. (1896); 28. 

 232, 468. (1898); 31. 17. (1900); 36. 209. (1902); 87.. 22. (1903); 88. 322. 

 (1904); 40. 529. (1905); 41. 478. (1906). 



) A. Schoenflies, Krystallsysteme und Krystallstruktur, Leipzig 1891), 

 p. 237; Zeits. f. Kryst. 20. 359. (1892); 54. 545. (1915); 55. 323. (1916). 



W. Barlow, Nature 29. 106, 205. (1883); Chem. News, 58. 3,16. Zeits. f. 

 Kryst. 23. 1. (1896); 25. 86. (1897); 27. 449. (1897); 29. 433. (1899). 



6) L. Wulff, Zeits. f. Kryst. 18. 503. (1888); 14. 552. (1888); E. Blasius, 

 Ber. d. bayr. Akad. d. Wiss. Miinchen 19. 47. (1889); Zeits. f. Kryst. 19. 512. 

 (1892); C. Viola, ibid. 81. 114. (1900); 85. 229. (1902); 41. 521. (1906); 

 A. Nold, ibid. 40. 13, 433. (1905); 41. 529. (1906); 48. 321. (1911); F. Haag, 

 Zeits. f. Kryst. 14. 501. (1888); K. Rohn, ibid. 85. 183. (1902); J. Becken- 

 kampi Zeits. f. Kryst. 44. 576. (1908); 45. 225. (1908); 47. 35. (1910); E. 

 Riecke, Zeits. f. Kryst. 36. 283. (1902). 



