ON THE STRUCTURE OF CRYSTALS. 311 



identical repetition of its parts is exhibited by any mechanical or geo- 

 metrical rigid system, this system being, in some of the cases, supposed 

 infinitely extended in every direction, a certain definite series or group of 

 correlated movements may be employed, each term of which is a movement 

 of such a nature that, while the system is actually shifted by it, the appear- 

 ance after the movement has taken place is absolutely unchanged, every 

 point moved being caused to travel to the place previously occupied by 

 some homologous point. ^ The fundamental condition that such a group 

 of movements may exist is that homologous parts everywhere bear an 

 identical relation to the system as a whole ; the members of the group 

 are so related that every individual movement may be regarded as the 

 resultant of some two or more movements also belonging to the group." 



While it is always found possible to partition any system of this kind, 

 in which the repetitions are continually repeated in every direction, in 

 such a way that the units obtained are all alike and sameivay-orientatp.d, 

 as in Bravais' systems,^ the latter property is, as has been said, but a 

 secondary one, and not of the nature of a definition, the condition stated 

 above constituting a definition complete in itself. A homogeneous struc- 

 ture can thus be classed according to the type of the infinite group of 

 coincidence movements which connect all its homologous parts. 



The obvious advantage of this method of dealing with homogeneity is 

 its complete generality — that it requires no further limitation of the 

 nature of the homogeneous structure than that which prescribes the kind 

 of repetition presented by its homologous parts.* Thus if molecules of a 

 certain individual symmetry with a relative space-lattice arrangement of 

 some kind are postulated, after the manner of Bravais and others, Jordan's 

 method, unlike Bravais', deals in one process both with the symmetry of 

 the individual, so far as this afiects the general symmetry, and also with, 

 the symmetry of arrangement. All possible molecular theories of crystals 

 can alike be subjected to Jordan's method, and it is independent of 

 them all. 



The following is the course of Jordan's argument : — After reminding 

 his readers that every movement of a solid body in space can be regarded 

 as a screw-spiral movement, he remarks that such a movement is fully 

 known when we are given — 



1. The situation in space of the axis of rotation A, which has also the 

 direction of translation. 



2. The angle T, through which the solid is turned about the axis. 



3. The longitudinal displacement t, to which the body is subjected in 

 the direction of the axis. 



He then observes that the displacement produced by two or more 

 such movements made successively can also be produced by a single screw- 

 spiral movement of some kind ; and the resultant of a number of move- 

 ments successively made can be definitely expressed in the terms just laid 

 down if the expi'essions for the component movements are known. 



Jordan next proceeds to point out that, a few movements being given, 

 it is possible to arrive at all the various movements or displacements 



' For a definition of a coincidence movement see Sohncke's Entivicliehmg elnei 

 Theoric der Krystallstruktnr, p. 28, or 3Iin. Mag., 1896, vol. xi. p. 125, note 3. Comp. 

 Schonflies, Krystallsydeme imd Krystalhtrudur, p. 54. 



- Schonflies, Krystallsysteme und Krijstalhtruciur, pp. 256 and 359. 



* See above, p. 306. 



* Cf. Schonflies, Krystalhysteme vnd Krystallstnictur, p. 44, par. 2. 



