78 



investigations it is perhaps preferable to give a simple demonstration 

 now, in which only the properties of the external, polyhedral form 

 of the crystals are made use of; we think this demonstration for 



the present purpose will be 

 sufficiently clear. *) 



Let ZO in fig. 86 be a sym- 

 metry-axis of the first order, 

 with a characteristic angle x> = 



217 



; ON is a possible 2 ) crystal- 

 edge, situated in the plane XOY 

 perpendicular to ZO. 



By rotations round ZO 

 through angles %, 2#, 3#, etc., 

 ON is repeated n times. Be- 

 cause all edges ON may be 

 used as coordinate-axes, we 

 shall here take OZ, ON, and 

 ON lt as Z-, Y-, and X-axis 

 respectively. If now CNNj_ be 

 a possible crystal-face 3 ), then 

 of course the same will be true 

 for CN 1 A^, CJV 2 N 3 , etc., and the mutual intersections of all these 

 planes, e.g. NC, N-^C, N Z C, etc., will be crystallographically possible 

 edges too. But if so, such planes as NCN 2> intersecting ON : in 5, 

 must be possible crystal-planes, because they pass through two 

 intersecting possible edges of the crystal. Therefore the plane 



Fig. 86. 



') A. Gadolin, Acta Soc. Scient. Fenn. (1871), 3; Ostw. Klass. d. ex. 

 Wiss. No. 75, p. 7, 7483. (1896). 



2 ) The intersections of possible (i.e. possible in the sense of Hauy's law) 

 crystal planes are always possible crystal-edges. Cf. the demonstration in: A. 

 Gadolin, Ostw. Klass. No. 75, p. 74 78. As a corollary it follows that every 

 plane passing through two non-parrallel possible edges of a crystal, is a pos- 

 sible crystal-plane too. 



3 ) If CAW] is not a possible plane, but e.g. CNn\, On { , being ^ ON t , 

 the successive intersections A7j, N^n 2> N%n s> etc., in plane XOY will not form 

 a closed polygon, if the lines A/w,, Af,w 2 , etc., be not continued until the}' 

 intersect in points s,, s.,, etc. The lines joining C with s, , s 2 , etc., are now the 

 intersections of a regular pyramid of n sides, and a figure analogous to the 

 one above, may now be used also for the purpose of demonstration. This last 

 one can therefore be considered to be sufficiently general. 



