310 EEPORT— 1901. 



deduction of the thirty -two classes directly on the existence of a homogeneous 

 molecular structure and not upon morphological considei-ations alone. 

 Yet it must be confessed that the various possible types of crystal sym- 

 metry were clearly and completely laid down by the morphologists without 

 any further speculation regarding structure than is necessitated by Haiiy's 

 law, and that every successive advance in the structure theories has been 

 guided or corrected by the knowledge so obtained. 



The Prmc'iple of Syimneirical BejJelition in Space. 



Shortly after the publication by Bravais of his elaborate and elegant 

 work, a new departure was made in the elucidation of homogeneity of 

 structure, the importance of which can scarcely be overrated. 



The first step was taken by Chr. Wiener,^ who laid down the principle 

 that regularity in the arrangement of identical atoms is presented when 

 every atom has the remaining atoms arranged about it in the same 

 manner ; ^ thus making homogeneity depend primarily on the continual 

 repetition throughout space of the same relation between an element and 

 the entire structure, regarded as unlimited, instead of laying stress on 

 sameway orientation.^ The principle adopted by Wiener, when employed 

 in all its generality, leads to an adequate classification, according to their 

 symmetry, of all cases of identical repetition throughout space whatever.'* 



The possibility of partitioning a homogeneous structure into similar 

 sameway-orientated parts whose centres form a parallelepipedal lattice '' 

 must always be the important property which enables us to trace to its 

 source Haiiy's great law of the rationality of indices ; but this possibility 

 is only a collateral fact when Wiener's principle is discussed ; indeed, the 

 carrying out of such a partitioning, while always possible,*" often compli- 

 cates instead of simplifying matters so far as the symmetry is concerned.^ 

 The problem to be solved, presented in its most general form, is not even 

 to find under what conditions the separation of the structure into similar 

 composite units of any sort can take place, but simply the analysis of the 

 nature of the repetition in space of the similar parts. 



Jo7'dan. 



Although Wiener made some interesting applications of his principle 

 and described several kinds of symmetrical repetition in space which are 

 examples of it, he did not deal with the subject exhaustively ; the solution 

 of the general problem was effected by Camille Jordan in a memoir the 

 title of which contains no reference to homogeneity or to crystals.*^ This 

 mathematician has furnished a perfectly general method of defining the 

 regular repetition in space of identical parts, and has shown that the typical 

 cases of such repetition are limited in number. He points out that, when 



Viola, ibid ,1S9G, vol. xxvi. p. 128, and IS'.iT, xxvii. pp. 399-40.5 ; De Souza-Rrandao, 

 Zeits. Kryst. 3Un., 1894, vol. sxiii. pp. 249-258, and 1897, vol. xxvii. pp. .545-555 ; 

 Barlow, Phil. Mari., 1901, series 6, vol. i. p. 3. 



' Die Grundziif/e der IVelturdnvng, Leipzig and Heidelberg, 1869. 



- ' Die Regelmas^igkeit lindet dann statt, wenn jedes Atom die anderen Atome in 

 iibereinstimmender Weise um sich gestellt hat,' ibid., p. 82. 



=• Cf. Mi7i. Mafl., 1896, vol. xi. p. 119 * See below, p. 321. 



^ Sohncke's Entwickelung einer Tlieorie der Krystallstruldvr, p. 207. 



^ Krystallsy Sterne und Krystallstrnctur, p. 360. Comp. Phil. Mag., 1901, series 6, 

 vol. i. p. 19. ' Comp. Mill. Mag., 1896, vol. xi. p. 125. 



^ ' Memoire sur les Groupes de Mouvements.' Annali di matetnatica jjura ed 

 a^pUcdta, Milapo, 1869, series 2, vol. ii. pp. 167 215, 322-345, 



