384 C. K. SWARTZ PROPOSED CLASSIFICATION OF CRYSTALS 



Again, the Orthorhombic, Monoclinic, and Triclinic crystals are divided 

 on a different basis from that employed in making the subdivisions of 

 dimetric crystals. The systems therefore appear to differ in rank. 



Relations of the rhomb ohedral and scalenohedral groups. — These 

 groups contain both a three-fold common and a six-fold alternating axis, 

 and hence may be referred, on the basis of period, to either the Trigonal 

 or Hexagonal system. While they have generally been referred to the 

 Trigonal system, it seems clear to the author that their natural place is in 

 the Hexagonal system, where he has placed them, for the following 

 reasons : 



The group containing the third order sphenoid of the Tetragonal sys- 

 tem is unquestionably four-fold. Dana makes it two-fold; Groth, four- 

 fold. Groth/s position seems correct. It possesses no center or planes of 

 symmetry, and it is impossible to develop it about an axis of two-fold 

 period. If the third order sphenoid, possessing four faces, is four-fold, 

 the analogous third order rhombohedron, with six faces, should have a 

 higher period, becoming six-fold. It is quite clear that the third order 

 rhombohedron bears the same relation to the Hexagonal system that the 

 third order sphenoid does to the Tetragonal system. The Tetragonal 

 character of the third order sphenoid is unquestioned. A consistent in- 

 terpretation would seem to refer the third order rhombohedron to the 

 Hexagonal system. 



Again, the Tetragonal and Hexagonal scalenohedra are the precise 

 analogues of the third order sphenoid and third order rhombohedron, and 

 their position is determined in the same manner. It seems fitting that 

 the analogous forms of the Tetragonal and Hexagonal systems should be 

 classified in the same manner. 



The third order rhombohedron can not be developed about a trigonal 

 axis without an appeal to a center of symmetry. It will be noticed that, 

 in all other forms, the results obtained by the use of a center of symmetry 

 spring directly from the axes and planes of symmetry as here employed. 

 It does not seem desirable to appeal to a center of symmetry in this case 

 only. 



Again, that the alternating axis is the dominant axis is indicated by 

 the fact that the highest occurring period, six-fold, is that of the alter- 

 nating axis, in harmony with the law of the Eationality of Parameters. 



It thus seems best to refer these groups to the Hexagonal system, giv- 

 ing a consistent development to Trigonal, Tetragonal, and Hexagonal sys- 

 tems and bringing out the evident close analogy between them. 



Center of symmetry. — The development employed has rendered it un- 

 necessary to use the center of symmetry as an independent element of 



