320 REPORT— 1901. 



but also, in the majority of cases, enantiomorphous similarity ; foi", wliik 

 in some few crystals the similar faces always bear an identical relation 

 to the whole, in most there are faces that occur in pairs (like a 

 right and left hand), the two individuals of which are enantiomorphously 

 not identically related to the crystal form. Unless this additional factor 

 of enantiomorphous similarity of parts be in some way introduced,^ 

 Jordan's method gives only the systems of repetition which belong to 

 one or other of the classes of crystal symmetry in which the similarity 

 is all identity, i.fi., only such as are enantiomorphs. This significant 

 fact is revealed in the work of the two inquirers, von Fedorow and 

 Schonflies, who established independently and simultaneously that a 

 definition of the symmetrical repetition of parts which includes enantio- 

 morphous similarity as well as identity of parts leads to types belonging 

 to all of the thirty-two classes of crystal symmetry.^ 



Pierre Curie '' shares with the two writers mentioned above the credit 

 of having established the general principles of repetition by which the 

 symmetry, whether of iinite figures, or of systems of figures, or of struc- 

 tures, may be completely investigated. He set himself to consider more 

 general arrangements of points than those dealt with by Bravais. These 

 points may be endowed with qualities independent of direction, such as 

 density, temperature, or with qualities requiring the most varied ideas of 

 direction and orientation, such as velocity, force, intensity of an electric 

 or magnetic field, intensity of power of rotation.^ (The homogeneous 

 arrangements thus obtained are not all ciystallographically possible, e.c/., 

 a sphere filled with a rotating liquid.'') There are two kinds of repe- 

 tition- one whicli leaves everything identically the same ■ as before 

 {de placements ind)jferents) and another in which the units of one part of 

 the system are the mirror-images of those of the other {sysihnes syme- 

 triques Vun de I'autre *>). Curie was the first to emphasise the necessity of 

 considering, in addition to ordinary axes and planes of symmetiy, axes 

 and planes of alternating symmetry {plans de symi'tric alterne, plans de 

 synietrie translatoire alterne '). Although the 2.30 classes of crystal struc- 

 ture obtained by Schonflies and Fedorow may be deduced from the prin- 

 ciples established in his papers, Curie limits himself to deriving the 

 thirty-two varieties of external form which are crystallographically pos- 

 sible.** 



Another writer of this date of whom mention should be here made is 

 B. Minnigerode, who arrived at the thirty-two classes of crystal systems 

 by means of the theory of groups and substitutions.'* 



' TLls is very clcaily brought out by Story-Mas-keiyne in his 3fDrj)JioIcf/i/ nf 

 Crystals, Oxford, 1895, p. 09, where the terms. ' metastrophic ' and ' antistiophic ' are 

 employed to distinguish the two sorts of relations. 



- The discovery of these thirty-two classes by the morphological crystallo' 

 graphers had in fact been due to the use of planes of symmetry ard centre of 

 symmetry as the basis of their reasoning; and these elements, of course, contain the 

 conception of enantiomorphous relationship 



^ 'Sur les questions d'ordre : Eepetilions,' Bull. Soc. jL'in., 18S4, vii. pp. 80-111 ; 

 'Sur la t^ymetrie,' ib., pp. 418-457. 



* Jh.,Tp. 89 * 7Z<.,p. 443. 



« Jb., p. 90. ' /*., p. 452. _ « lb., p. 454. 



' ' Untersuchungen iiber die Symmetiieverhjiltnisse und die Elasticitiit der Ivrys- 

 talle,' KacJir. d. It. Ges. d. M'iss., Gottingen, 1884, pp. 195-226, 374-384, 488-492 ; 

 ' Untersuchungen iiber die SymmetrieverLultnisiie der Krystalle,' Kcves Jahrb,, 1887 ; 

 Beilago., Bd. v. pp. 145-166. 



