ON THE STRUCTURE OF CRYSTALS. 323 



5. Similarly, if T be transposed into T' by the operation of a plane of 

 gliding symmetry, T is also so transposed by the operation of a simple 

 plane of reflection having the same situation. 



6. Hence, corresponding to any given group of operations containing 

 screw axes or planes of gliding symmetry, there exists another group of 

 operations which effect the same changes of direction, but whose elements 

 of symmetry are axes of rotation or planes of reflection, and these are 

 such as belong to a space-lattice. 



7. From this it follows that in the groups of space-operations the 

 only axes found are those of the orders characteristic of space-lattices, 

 i.e., digonal, trigonal, tetragonal, and hexagonal axes. 



The relations between groups of space-operations {Raumgru2Jpen) of 

 different types can be traced by means of the similar relations subsisting 

 between allied (' isomorphous ') types of symmetrical operations effected 

 solely about a single point or ' centre ' (Punktgruppen) ; ^ the latter, 

 since the kinds of axes admissible are limited as above, are those which 

 characterise the centred forms of the thirty-two types of crystal sym- 

 metry. 



Two operations are termed by Schonflies ' isomorphous ' when their 

 planes and axes of repetition have the same directions and the angles of 

 rotation of the latter are the same. 



A group of space-operations and a group of centred operations are 

 termed isomorphous when every operation of the former is isomorphous 

 with an operation of the latter. 



By this method of comparison it is shown that every one of the groups 

 of ' space-operations ' involves the general symmetry which governs the 

 symmetry of repetition of like directions in one or other of the thirty-two 

 classes of crystal symmetry. 



The mirror-image of a screw movement is a similar movement of the 

 opposite hand. Among the groups of operations corresponding to 

 Sohncke's sixty-five systems which contain screw movements, only 

 such as possess sci-ew-axes of two opposite hands can be utilised for 

 the purpose of deriving groups of space-operations containing mirror- 

 image repetition : such are (1) those which contain screw-axes whose 

 translation component is equal to a half -translation ; - (2) those which 

 contain for each screw-motion in one direction an equal screw-motion in 

 the opposite direction. 



By applying the above principles Schonflies is able to show that the 

 sixty-five systems of Sohncke are increased to 230 groups of operations, 

 all of which, from what has been said, must belong to one or other of the 

 thirty-two types of crystal symmetry. 



A complete set of similar plane-directions may be drawn in a 

 Schonflies group of operations, in a way similar to that already indicated 

 for finding identical planes in one of Jordan's infinite groups of move- 

 ments.^ Thus : — 



^ KryHaltsijsteme und KrystalUtfitduf, pp. 330* 301, 374, 378, 8S3. 

 _ - This case is illustrated by fig. 5, in which the translation component of the 

 axis C (necessary to derive e/ from c,) is one half of the translation ^, o," belonging 

 to the system. Successive points may be regarded as lying either on a right-handed 

 spiral (as e„ c„' c,") or on a left-handed spiral|(a3 c^ c^ c"). 



^ See above, p. 312. 



y2 



