83(3 REPORT— 1901. 



packed spheres, either equal or of two or three different sizes (reprs^ 

 senting the atoms), may be constructed so as to possess the symmetry of 

 many holohedral, hemilaedral, or tetartohedral crystals, and to be in 

 harmony with their physical properties. ' 



Among recent writers mention should be made of C. Viola. He has 

 employed the method of quarternions to derive the thirty-two classes of 

 crystal symmetry ^ and has also given an elementary exposition of these 

 classes based upon the planes of symmetry.^ In a recent paper ^ he 

 questions the ultimate validity of the law of rational indices. 



It is sufficient liere to point out that all the systems devised by Kelvin, 

 Barlow, Turner, Sollas and others, being homogeneous arrangements, 

 must correspond geometrically to one or other of the 230 types of 

 iSchonflies and Fedorow, and must all, as to their symmetry, be ultimately 

 reducible to a certain number of interpenetrating space-lattices. As they 

 go beyond the geometry of the subject their consideration is postponed 

 for the present. 



Summary. 



With the establishment of the 230 tjpes of structure the purely 

 geometrical study of the problem seems to have attained something like 

 hnality. The history of its development, as sketched above, is the history 

 of an attempt to express geometrically the physical properties of crystals, 

 and at each stage of the progress an appeal to their known morphological 

 properties has driven the geometrician to widen the scope of his inquiry 

 and to enlarge his detinition of homogeneity in order that it may include 

 types of symmetry which did not fall within the more restricted defini- 

 tion. The necessity of explaining liemihedrism led to the system of 

 Sohncke ; the necessity of accounting for the known symmetry of dioptase 

 led to the further extension of Sohncke's principles. 



The two most satisfactory features of the tinal geometrical solution of 

 the problem are the following : (1) A single jDrinciple — namely homo- 

 geneity according to the wider definition — is sufficient to account for the 

 two leading characteristics of crystals, their a?o!otropism, and the law of 

 rational indices. (2) The lines are now laid down within which specula- 

 tion concerning the actual structure of any crystallised substance can 

 range. 



There are three problems to be solved in explaining the structure of 

 crystals: (1) What are the parts of which a crystal consists? (2) How 

 are they arranged ? (3) Why are they arranged in this particular 

 way ? 



We have now good reason to believe that a partial answer has been 

 found to the second question, and that whatever may be the parts of 

 which a crystal consists they must be arranged according to one or other 

 of the 230 types of symmetry ; Sohncke systems and Bravais space- 

 lattices are, of course, special cases of these. 



' Compare also A. Turner, Das Prohlem der Kryatallisation, Leipzig, 1897, and 

 W. J. Sollas ' On the Intimate Structure of Crystals,' Proc. Boy. Sue, 1898, Ixiii. 

 pp. 270-BOO; I'JOl, Ixvii. pp. 493-496. 



- Ueber die Symreetrie der Krystalle und Anwendung der Quaternionrechnung, 

 Kenes Jalirb., 1896, B-ilage Bd. s. pp. 49.5-532. 



^ Elementare Dar^tellung der 32 Krystallklafse, Zeits. A'ri/sf. Min., 1897, xxvii, 

 pp. 1-40. 



* Zur Begiiindung der Krystallsymmetrjen, ■'lid., 1901, sxxiv, pp. B53-388, 



