80 



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. l ) 



Let ZO in fig. 88 be a sym'- 

 metry-axis of the first order, 

 with a characteristic angle a = 



r\ 



' ; ON is a possible 2 ) crystal- 

 edge, situated in the plane XOY 

 perpendicular to ZO. 



By rotations round ZO 

 through angles a, 2a, 3a, 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 CNN^^ be 



a possible crystal-face 3 ), then of course the same will be true for 

 CN 1 N 2t CN 2 N 3 , etc., and the mutual intersections of all these planes, 

 e.g. NC, Nf, N 2 C, etc., will, of course, also be crystallographical- 

 ly possible edges. But if so, such planes as NCN 2 , intersecting 

 ON l in 5, must be possible crystal-planes, because they pass through 

 two intersecting possible edges of the crystal. Therefore, the plane 



1 ) A. Gadolin, Acta Soc. Sclent. 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 of this 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 also a pos- 

 sible crystal-plane. 



3 ) If CNN 1 is not a possible plane, but *e. g. CNn v On v being ^ ON V 

 the successive intersections Nn v N t n z , N z n 3 , etc., in the plane XOY will not form 

 a closed polygon, if the lines Nn v N^n^, etc., be not continued until they 

 intersect in points Sj, s z , etc. The lines joining C with Sj, 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. 



