to study of the Structure of Crystals. 



129 



P is anij point xyz. The other equivalent points F^ {xijz), P^ 

 [xtjz) and P3 (5yJ) result from P by the forenamed operations'^ 

 Similarly the equivalent points of' each of the other thirty-one 

 point groups can be obtained from their characteristics of sym 

 metry. The five point groups having cubic symuietry are 



SvmboP2 Class of Symmetry No. of Equiya- 



Schonflies, Dana, Groth lent Points 



^ _, . Pentagonal 

 1. T Tetartohedry Tetartohedral dodecahedral ^'^ 



'^- Th WiSy''' Pyritohedral Dyakistetrahedral 24 



3. T,i 1 emimorp 1 q^e^rabedral Hexakistetrahedral 24 

 ^^ nemihedry 



^ Enantimorphic -^, • 1 1 , Pentagonal 

 ^- ^ hemihedry Plagiohedral ieositetrahedral ^^ 



5. Oh Holohedry Normal Hexakisoctahedral 48 



An extended arrangement of points in space which will 

 have the symmetry of one of the crystal classes can be 

 obtained by arranging these point groups according to 

 some regular pattern which will itself have the total 

 symmetry of the crystal system; such an extended 

 arrangement of points in space is a space group. All of 

 the centers of the point groups arranged according to 

 one of these regular patterns are points of what is termed 

 a space lattice. Fourteen space lattices have been shown 

 to be crystallographically possible : one underlying all 

 triclinic crystals, two possible monoclinic lattices, four 

 orthorhombic, two tetragonal, two hexagonal, and three 

 cubic. If the principal axes of synnnetry of the lattice 

 are taken as coordinate axes, each lattice can be defined 

 by giving the translations along these axes which, applied 

 to one point of the lattice, will give every other. The 

 three cubic lattices are 



"x is written for -x, etc. 



^^ Unless otherwise noted, the notation that will be used is precisely that 

 of Sehonflies. Point grouj)s are designated by capital letters, either alone or 

 with subscripted numerals and elevated small letters. A particular case, to 

 be discussed in detail shortly, will perhaps make the reason for this notation 



clear. The holohedry of the monoclinic system (C;^) is one of the cyclic 



groups, its principal axis of symmetry is tivo-fold and it has a horizontal 

 plane of s^anmetry. In other words, the capital letter refers to the type of 

 group, the numeral to the nature of the principal axis, and the small letter 

 (except i for inversion) to the position of the i)rincipal plane of symmetry. 

 In an analogous fashion the space groups that are isomorphous with this 

 point group are written Cgh', C2h-, and so on. 



Am. Jour. Sci. — Fifth Series. V'ol. I, No. 2. — February, 1921. 

 9 



