322 REPORT— 1901. 



movements. He thus constructs composite groups &£ operations which 

 act throughout space, but comprise, in addition to Jordan's groups, cer- 

 tain mirror-image operations with respect to series of parallel planes or to 

 systems of centres of inversion.' He calls the groups of operations, 

 whether those of Jordan or those added by himself, ' space-groups ' (Eauin- 

 gruj}pen).^ 



As in the case of Jordan's groups of movements, the symmetry of 

 any given group is rendered easier to trace if a number of similar par- 

 ticles or bodies are placed in all the positions, throughout some consider- 

 able space, in which they would be located by applying all the operations 

 of the group to some particular body. In order to accomplish this, in 

 the groups which contain mirror-image operations similar right-handed 

 and left-handed bodies will have to be employed in equal numbers.^ 



It may be maintained that the likeness of parts thus defined by 

 Schonflies, involving as it does two distinct sorts of resemblance — 

 identity and enantiomorphous (or mirror-image) similarity — should 

 scarcely be called, when taken collectively, homogeneity of structure ; 

 it would be well, perhaps, if it could be expressed by some new word of 

 wider significance. 



Generation of the Various Groups of Operations (Raumgruppen). 



Schonflies employs a symbolic method in order to deduce the various 

 types of possible groups of operations. 



The following propositions indicate briefly the method pursued by 

 him, without introducing his symbols : — 



1. Only such of Jordan's groups of movements as contain a group of 

 translations which all bear finite (and not infinitesimal) relations to one 

 another, and are, therefore, capable of producing a space-lattice {Rmim- 

 qitter), can obey the law of rational indices ; and are, therefore, available 

 for the crystallographer.* It is only to these groups that Schonflies 

 applies mirror-image operations.'^ 



2. The complete set of translations thus forming part of a Schonflies 

 cfroup of operations must be brought to coincidence with itself {Deckuny) 

 by every other operation of the group.'' 



3. In addition to planes of symmetry, simple axes of symmetry, and 

 the screw-axes of Sohncke, Schonflies (like Curie) introduces ' planes of 

 ^fliding symmetry ' (Gleifebeneny as another possible mode of repeti- 

 tion that can be employed in a group of space-operations. A plane of 

 gliding symmetry is the result of combining reflection over a plane 

 with a translation parallel to that plane. 



4. If a given translation, T, be transposed by the operation of a screw 

 axis into another translation, T', T is also thus transposed by the opera- 

 tion of a simp)le axis of symmetry having the same situation and angle of 

 rotation. 



' Schonflies, Krystallsystcme und Krystallstructur, pp. 334 and 55G, 

 " lb., p. 359. ^ See 3Im. Mag., 1896, vol. xi. p. 119, and see below, p. .333- 



" Krystalhystcmc iind Krystallstructur, pp. 360, G36. '- lb , pp, 3()0, 3G1. 



" lb., p. 362. Schonflies calls sub-groups of operations which have thia property 



ausyezeicJmete Utitergruppeii. 



' lb., p. 367. Schonflies calls that one of the various possible movements about 



a particular axis which has the smallest angle of rotation and the smallest positive 



translation the ' reduced movement ' (rcducirte Bewegung), 



