312 REPORT— 1901. 



obtainable by combining these given movements executed successively any 

 number of times in any order wliatever. Of groiqys of movements arrived 

 at in this way, some are of a finite character, and some contain movements 

 infinitely small ; the remaining kind — those which consist of movements 

 whose loci extend infinitely throughout space in every direction, and 

 which are none of them infinitely small as compared with the others — 

 comprise, as was subsequently perceived,' all those that are available for 

 the production or definition of homogeneous structures which display the 

 symmetry of cr3'stals.^ 



The movements belonging to an infinite group of movements, like any 

 individual movement, can be completely defined by reference to certain 

 axes of rotation and directions of translation ; but for the sake of per- 

 spicuity it is desirable to place a number of similar particles or bodies in 

 all the positions, throughout some considerable space, to which one of 

 them would be moved by the various movements constituting the group. 

 When this is done the kind of symmetry presented by the system formed 

 of the group of movements can be readily perceived,^ and at the same 

 time the nature of the parts repeated can be left an open question. 



If it be desired by the crystallographer to find in a given homogeneous 

 system a complete set of identical planes by means of the group of move- 

 ments proper to the system, the following course may be adopted. 



Take three points — A, B, C — whose identical relation to the system 

 is such that the aspect of the unlimited structure is the same and jweseids 

 the same orientation viewed from each of them, and let their distances 

 apart be not great as compared with the minimum distances separating 

 homologous parts of the structure. The repeated carrying out of the 

 three translations — AB, BC, CA in both directions — will locate an infini- 

 tude of points lying in the plane of the three points, and all having 

 precisely the same relation to the structure as that presented for the latter. 

 This plane may therefore be designated a homogeneous 2}^ane,'^ and since 

 the translations of the structure are not infinitesimal, it is easy to prove 

 that a plane so situated will obey the law of the rationality of indices 

 when referred to axes which pass through strings of identical points.* 

 When such a plane is subjected to the various coincidence-movements 

 constituting the group characteristic of the structure, an infinite set of 

 planes is found, which all have an identical relation to the structure. 

 The number of different orientations presented by the planes is limited. 



SoJincke. 



The treatment of homogeneity of structure by Jordan's method leads 

 to a classification which discriminates the various types of identical 



' See below, p. 315. Cf. Erijstallsystevic it. Krystallstructur, pp. 360 and G3G ; 

 also see above, note 3, p. 298. 



- It is interesting to notice that Jordan does not appear to have regarded his 

 work as throwing any fresh light on crystal structure, but treats Bravais' work as 

 complete in this direction. He says: 'M. Bravais has studied this question; the 

 particular cases which he has discussed, and of which he has made a remarkable 

 application to crystallography, are the most important. Nevertheless I believe 

 there is at the present time some interest in treating the problem quite generally.' 

 (^Memoire sur les Grovpes de JMovrementK, p. 168.) 



= See Mm. Mag., 1896, xi. p. 113, and see below, p. 333. 



* See Phil. Mag., 1901, series G, i. p. 19. 



* The hypothesis wjth regard to crystals is that their faces lie in homogeneons 

 planes. See Bravais, Etudes Crystallogra^higues, p. 103, 



