144 



the axial ratio of this hexagonal arrangement is:a:c = l:2v r f = 

 1 : 1,6330, or half of it, = 1 : 0,8165. This value is therefore 

 descriptive l ) 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 wellknown 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 

 both theoretical values, appears to be only an approximate one: 

 for magnesium it is: 1 : 1,6242; for beryllium: 1 : 1,5802; for 

 arsenic: 1 : 1,4025, while for most of them those values oscillate 

 round 1 : 1,33. 



The agreement is better in the case of the oxides and sulphides 

 of bivalent elements as zinc, cadmium, beryllium, etc., where evidently 

 the same conditions must exist. A few examples will make this clear 2 ) : 



Zinc-oxide a : c = 1 : 1,6077. 



Zinc-sulphide a : c 1 : 0,8175. 



Beryllium-oxide a : c = \ : 1,6305. 



Cadmium-sulphide a : c = \ : 0,8109. 



etc. 



18. Notwithstanding this obvious, at least partial, agreement 

 between measurements and calculations, we shall see that the struc- 

 tures following from the above mentioned theory agree only occa- 

 sionally with the result of the recent investigations of some of 

 these substances by means of Ron t gen-rays. It can hardly be 

 denied that serious objections have occasionally been raised against 

 the theory mentioned, against its fundamental suppositions, and 

 against the conclusions drawn from them, - - suggestive as they 



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

 such, the ratio: a : c = 1 :~[/^2 = l : 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. 



2 ) C. Friedel, Compt. rend, de 1'Acad. d. Sciences Paris, 52, 983, (1861); 

 O. Miigge, Jahrb. f. Miner., 2, 1, (1882); E. Mallard, Ann., des Mines, 12, 

 427, (1887); H. Traube, Jahrb. f. Miner., Beil. End. 9, 147, (1894). 



