140 



course being also perpendicular to the symmetry-planes mentioned 

 above. 



The screw-axes do not pass through any sphere-centres at all; but 

 three planes of symmetry pass through every senary screw-axis and 

 through the centres of the three nearest spheres of consecutive layers. 

 Three double sets of binary axes lie in planes midway between the 

 layers of most closely packed triangularly arranged spheres, and per- 

 pendicular to the last described symmetry-planes, and they intersect 



the senary screw-axes. Through every 

 pair of nearest senary screw-axes a 

 plane of "gliding" symmetry can be 

 brought, which planes are evidently 

 perpendicular to the binary axes just 

 mentioned. In planes midway between 

 the consecutive layers, the symmetry- 

 centres of the endless system are 

 situated on every senary screw-axis 

 and at points midway between them. 

 If the value of the translation perpen- 

 dicular to each layer, by which a sphe- 

 re of the first layer can be brought to 

 coincidence with a superposed sphere 



Fig. 122. 



Hexagonal Assemblage of Equal 

 Spheres. 



of the third layer, be taken as the parameter of the c-axis, while 

 the distance of two contiguous spheres in each layer is taken as 

 a-axis, - - it will be obvious that the axial ratio of this hexagonal 

 arrangement is:a:c=l:2l/'--=l: 1,6330, or half of it, = 

 1 : 0,8165. This value is therefore descriptive x ) of such most closely 

 packed hexagonal assemblages of equal spheres. 



First now, there are a number of chemical elements which crys- 

 tallise in the cubic system: silver, mercury, gold, copper, several 

 platinum metals, etc., are well-known examples of this. 



Secondly, a number of elements are hexagonal: magnesium, beryl- 

 lium, arsenic, etc., may be mentioned among others. Moreover in 

 cases of dimorphism of such elements, the change of cubic into hexa- 

 gonal symmetry, and vice-versa, is frequently observed. 



However the agreement of the axial ratios of these elements with 



!) If however the perpendicular to the now adopted a-axis were chosen as 

 such, the ratio: a : c -= 1 : ]/2 = 1 : 1,4142 would have been the descriptive 

 value for this assemblage, which of course is equally appropriate for the purpose 

 of characterising the hexagonal assemblage under consideration. 



