328 REPORT— 1901. 



Six planes of reflexion pass through the hexagonal axis, H1H2, making 

 angles of 30° with one another (such are the planes HoH^Hj, H,2Hf;H5, 

 and HoHgHj), and, parallel to these planes, there must be throughout 

 the structure a series of equidistant planes of reflexion. 



Further, there is a centre of symmetry on the hexagonal axis ; we may 

 suppose it to coincide with H,, since this was arbitrarily chosen. All 

 the nodes of the Bravais-system are such centres of symmetry, and 

 in addition all the middle points of the ro\sfS, i.e., all, the points H, D, 

 and O. 



Further, the presence of this centre combined with the hexagonal axis 

 necessitates the existence of planes of reflexion perpendicular to the axis 

 and passing through the centres, and, consequently, separated from one 

 another by 0|H|, half the parameter, HqH,, of the axis. Also, perpen- 

 dicular to each of the planes passing through the hexagonal axis there are 

 a series of diagonal axes of rotation passing through the centres of 

 symmetry lying on these planes. 



Such, then, are, in the case in question, the elements of crystalline 

 symmetry which fill space. 



The elementary cell is easily determined, since the elements of 

 symmetry must lie on its surface ; it is the right jDrism with triangular 

 base 0,0.2TH|D|T|, which has its bases in two principal planes ; its 

 edges are a hexagonal axis H,0,, a trigonal axis T,T._„ and a digonal axis 

 D,02 ; its side faces are three planes of symmetry ; the four corners 

 H,, D|, O2, Oi, are centres of symmetry, but the corners TT,, situated on 

 the trigonal axes, are not centres. 



To obtain the complex cell we must apply to the fundamental cell the 

 appropriate elements of symmetry — i.e., in this case the hexagonal axis 

 and two planes of symmetry, 0|OoH,D, and HiD|T| — whence we obtain 

 a right prism with hexagonal base whose edges are the trigonal axes, i.e., 

 the cell of fig. 3. 



(By taking another set of the primary elements of symmetry anotlicr 

 complex cell will be obtained.) 



A corresponding crystalline structure will be obtained by furnishing 

 each elementary cell in a similar manner with contents of any nature. 



Feclorotc. 



As has been said above, the 230 types of crystal structure were inde- 

 pendently established and investigated by E. von Fedorow. 



The researches of this author which relate to the subject of crystal 

 structure begin in the year 1885 with a. general treatise on the 'Theory 

 of Figures,' published (in Russian) with copious illustrations in the ' Trans- 

 actions of the Russian Mineralogical Society,' xxi. pp. 1-279 : this was 

 followed in 1888 by a memoir on the ' Symmetry of Finite Figures,' pub- 

 lished (in Russian) in the same journal, xxv. pp. 1-52, and by one on the 

 ' Symmetry of Regular Systems of Figures,' published (in Russian) in 

 1890. 



The above are not only among the earliest treatises on these subjects, 

 but they contain also almost all that is essential in the author's later 

 development of it, and some results that have been independently pub- 

 lished by other investigators to whom his Russian papers were not known. 

 An abstract of some of the early papers was given by Wulfi' ^ and by 

 Fedorow himself.- 



' ^eits. Kryst- M%n., 1899, vol x?ii. p. 610, ^ 76., 1893, vol. xxi, p, 679, 



