ELEMENTAKY DEVELOPMENT 383 



no three-fold period and the Amebahedral class no period less than four. 

 Tn addition to the preceding we may also have one-fold Axial and one- 

 fold Hedral forms. 



Thirty-two groups are developed in this manner. They are summa- 

 rized in the following table, the periods possible in each class being given 

 below its name. 



Singly Doubly terminated 



terminated 



Direct Oblique Alternate 



Axial Axial Orthoaxial Polvaxial Amebaxial 



1, 2, 3, 4, 6 2, 3, 4, 6 2, 3, 4, 6, 2, 4 2, - 4, 6 



Hedral Hedral Orthohedral Amebahedral 



1,2,3,4,6 2,3,4,6,2,4 4,6,4 



The reference of the groups to the various systems may then be devel- 

 oped as outlined in the preceding paragraphs. 



INFERENCES 



It may be of interest to give certain deductions springing from the pre- 

 ceding discussion. 



Relative rank of classes and systems. — The classes appear to be more 

 natural in character than the systems. They are certainly less artificial 

 than the systems, if the latter are made, as has been generally the case, to 

 depend upon the crystallographic axes, which are imaginary lines sub- 

 jectively present in the mind of the student. This is well seen in the 

 difference in usage in the Trigonal system, where the axes of Miller or 

 Bravais can be employed equally well. It is also quite possible to refer 

 the Hexagonal and Trigonal forms to orthorhombic axes, as frequently 

 done by Barlow and Pope in their recent paper, "The relation of crys- 

 talline form and chemical constitution." 5 The classes, on the contrary, 

 depend upon the axes and planes of symmetry objectively present in the 

 crystal, and are hence natural units. 



Relative rank of the systems. — The various systems do not appear to be 

 coordinate in value, but represent divisions of different rank. Thus the 

 Isometric system is coordinate with the Dimetric division comprising 

 crystals of the Trigonal, Tetragonal, and Hexagonal systems, and also 

 with the Trimetric division. This is also shown by the optical proper- 

 ties, in which respect the Isometric system corresponds to the "Uniaxial 

 and Biaxial divisions of crystals. Like those divisions, it may be sub- 

 divided on the basis of period, yielding two-fold and four-fold sections. 



5 Journal of the Transactions of the Chemical Society, vol. 91, 1907, part 2, pp. 1150- 

 1214, and vol. 94, 1908, p. 1528. 



