SUMMARY 397 



Notes on Table IIS 5 



Unless otherwise stated, the formulae preceding the period (.) indicate the 

 divisions given in the foregoing discussion, which are summarized in table I. 

 The number following the period shows the author's number of the group. 

 Letters are used when the author has so employed them. § indicates section 

 in which the group is discussed. 



Hessel. — The letters indicate character of chief axis; for example, u = un- 

 gleichendig, etcetera. The large number shows number of such axes ; the first 

 small number indicates whether axis is single (1) or double (2) ; the second 

 small number shows period of the axis. This l 2 G 2 signifies one double axis, 

 gleichstellig, 2-fold." 



Miers. — Letter signifies division; for example, p = polar; a = alternate, 

 etcetera. Small figure 2 signifies double (di) form. 



Groth. — Numbers indicate group. Physikalische Krystallographie, 4th edi- 

 tion, 1895, pp. 329-331. 



Summary 



The purpose of the preceding discussion is to point out the existence of 

 certain natural and well defined types of symmetry in crystals, and to 

 present a development of the thirty-two groups which is adapted to the 

 use of elementary students and which emphasizes simple and natural 

 relations of crystals springing from the existence of these types. The 

 subject is treated in two parts. 



Part I contains the discussion of the classification proposed by the 

 author. 



1. An elementary development of the thirty- two groups is given, by 

 which it is shown that all crystals fall into seven fundamental classes of 

 symmetry. 



2. The seven classes are named, the characteristics of each are dis- 

 cussed, and the term class defined. 



3. The various classes and groups are referred to the accepted systems. 



4. It is shown that this development expresses the larger relations and 

 harmonizes with the fundamental physical properties of crystals. 



45 Tbis table is based in part on a similar table of Schoenflies, "Krystall Systeme und 

 Krystall Structur," 1901, table iii, p. 104. 



48 Tbese formula? were first used by Hessel in a later publication, "Ueber gewisse 

 merkwiirdige statiscbe und mechaniscne Eigenscbaften des Raumes," Marburg, 1862. 

 Universitatsscbrift. 



