ON THE STRUCTURE OP CRYSTALS. 325 



group, as compared witli the group from which it was obtained, will in 

 different cases present — 



a. A greater number of orientations of the planes belonging to the 

 derived system. 



b. The same number. In this case the change will consist solely in the 

 increased closeness of the planes of a set, and the type among the 230, 

 which is exhibited, will sometimes be diff'erent, sometimes the same. 



The converse proposition is — - 



2. The withdrawal of some operations from a group, entailing the 

 symmetrical omission of some of the sets of parallel planes, or of some of 

 the planes in each set, leads to the derivation of a distinct group of 

 operations. There will in different cases be — 



o. Fewer directions of orientation for the planes in the derived group. 



/3. The same number of directions, associated in some cases with the 

 preservation of the same type, in some cases with the development of a 

 different type among the 230. 



As a simple example of the application of the principles established 

 above consider the hemimorphous class of the monoclinic system. ^ It 

 possesses an axis of two-fold symmetry, which in the space-group may 

 appear as an axis of rotation or as a screw-axis. Now, in the monoclinic 

 system there are two lattices : one rhomboidal and the other composed of 

 rhomboidal prisms with centred faces. We obtain two groups from the 

 former by combining it with an axis of rotation, and with a screw axis ; 

 from the latter we obtain only one group, since in this case the same 

 group is derived by the addition of either set of axes. 



Like Jordan's groups, those traced by Schonflies are really mei'e 

 groups of geometrical processes, independent of the nature of the material 

 system concerned ; but it is convenient to regard the processes as applied 

 to something more tangible. Schonflies himself supplies this want by 

 introducing the conception of atomic structure, and of its definite par- 

 titioning. Here the reader must beware lest the nature or configuration 

 of the atoms or particles themselves be confounded with the nature and 

 distribution of the structure considered with respect to them, and lest the 

 possibilities of mere geometrical partitioning be confounded with those of 

 a partitioning into conceivable physical units.^ 



Schonflies treats his work of discriminating 230 types of groups of 

 operations (Raumgruppen) as preliminary to a direct application of his 

 results to a molecular theory of matter, which he sets before himself from 

 the outset ; the reader might, therefore, suppose that the existence of 

 molecules with void spaces between them is essential in order that the 

 geometrical derivation of the 230 types may be applicable to crystals.-* 

 Thus Schonflies says : ' By a regular assemblage of molecules of unlimited 

 extent is understood a molecular assemblage infinitely extended in all 

 directions, which consists entirely of similar molecules, and possesses the 

 property that around every molecule the disposition of the infinite system 

 formed by the other molecules is similar.' ^ And a little later he lays 



system von G, in sich uberflilirt, so kann jede zu G isomorphe Raumgruppe durch 

 Multiplication einer zu G, isomorphen Groppe r, mit einer zu C isomorphen Opera- 

 tion £ erzeugb warden, vorausgesetzt, dass 2 eine Deckoperation f iir die Axen von r, ist.' 



' KryitalUystenie und Krystallstr^tctur, p. 406. 



= Cf. Min. Mag., 1896, vol. xi. p. 129. 



' Krystalhysteme vnd KrystaUstrvctur, p. 237- * Jhid., p. 239. 



