WORK OF MINNIGERODE AND OF SCHOENFLIES 393 



&. Two-fold period and center of symmetry = four-fold period. 

 Groups 23-24. 

 3. Possessing two-fold period, with or without center of symmetry = 

 two-fold period or without axis of symmetry. 



a. Orthorhombic. Groups 25-27. 



b. Monoclinic. Groups 28-30. 



c. Triclinic. Groups 31-32. 



Thirty-two groups of crystals are obtained by employing the periods 

 possible in crystals. 



Minnigerode's results, though obtained by an independent method, 

 coincide fully with those of the preceding workers. His discussion, while 

 highly mathematical, is elegant and brief. 



Schoenflies. — In the year 1891 A. Schoenflies published the most im- 

 portant contribution to the classification of crystals (save perhaps that of 

 Fedorow) since the memoire of Gadolin. His work, entitled "Krystall 

 Systeme and Krystall Structure," 39 is devoted to the discussion of the 

 geometrical form and inner structure of crystals. 



Like Gadolin, he restricts his discussion to the forms possible in crys- 

 tals under the law of the Eationality of Parameters. He recognizes sym- 

 metry of two kinds: 40 



A. Symmetry by rotation, which he terms symmetry of the first kind. 



B. Symmetry by reflection (variously combined with rotation), termed 

 symmetry of the second kind. 



Forms of the first kind possess an axis of symmetry only. Forms of 

 the second kind are produced by passing planes in various positions 

 through the axes of the first kind. The following table 41 shows his 

 divisions : 



I. Possessing axes of symmetry only. II. Possessing planes combined with 



1. No axis. Identity. C v axes of symmetry. 



2. One axis. Cyclic. Cn. Cyclic. 



Plane horizontal. C h . 

 Plane vertical. C v . 



3. Several axes, one principal. v - ( Plane horizontal. V h . 



Period 2, Vierer. V. vierer I Plane diagonal. V d . 



Period over 2, Dihedral. Dn. Dihedral ....{ f^ ^ona^ ' D* 



4. Several axes of period over 2. 



Period 2, Tetrahedral. T. Telrahedral. 



Plane horizontal. T h . 

 Plane diagonal. T d . 



Period 4, Octahedral. O. Octahedral . . Plane horizontal. O h . 



These divisions develop thirty-two groups of symmetry by changes in 

 the periods of the axes. 



39 Leipzig, 1891. See an excellent brief synopsis of his results in G. H. Williams' 

 Crystallography, third edition, 1892, pp. 183-195. 



40 Ibid., p. 129. 



« Ibid., pp. 74, 102. 



