Mr. W. Barlow on Crystal Symmetry. 9 



them will cut some other space-unit in the same manner. 

 Thus, in the plane example just given, differently directed 

 lines A B, C l3, E F cut three different space-units, K, L, M, 

 and indeed the ultimate structure, identically . 



Now, as the space-units are all alike, and bear the same 

 relation to the structure, every one of them can have a line 

 drawn to intersect it in the same manner ; those of such 

 lines whose similarly-traversed space-units are similarly 

 orientated will be parallel to one another, but in differently- 

 orientated space-units the corresponding lines will, it is 

 evident, be differently directed, unless indeed the direction of 

 the latter is exceptional so as to be similarly related to two or 

 more differently -orientated space-units. Therefore the pro- 

 position is established. 



It has been said that homogeneous structures are of the 

 same class when the number and arrangement of like direc- 

 tions is the same in them. It is evident from the foregoing 

 that where the likeness of the directions amounts to identity, 

 this is equivalent to saying that structures will be of the 

 same class when the various orientations of their space-units 

 display the same kind of relative arrangement. Consequently 

 to tind what variety of classes is possible, the task which lies 

 immediately before us is to ascertain what various kinds of 

 relative arrangement of the different orientations of the space- 

 units are presented when all possible types of homogeneous 

 structures, which are so constituted that their space-units 

 are of finite dimensions, are examined. 



The second important property of such homogeneous 

 structures furnishes the next step towards this ; it is as 

 follows : — 



11. tor every homogeneous structure a group of coincidence- 

 movements exists which comprises every movement of the entire 

 mass requisite to carry every space-unit to the identical place 

 occupied originally by any other space-unit, and every rigid 

 line drawn to intersect the mass to the place of some identically 

 related right line ; the ultimate structure viewed from any 

 fixed point presenting precisely the same aspect after as before 

 each of these movements*. 



This property is an evident consequence of the similar 

 nature and situation of the identical space-units. 



Now, according to a theorem of Chasles, any rigid system 

 can, by means of some screw-spiral movement, be carried 

 from any initial position to any other given final position. 



* See "Memoire sur les groupes de monvements," par C. Jordan: 

 Annali di matematica pura ed applicata, Serie 2, Tomo ii. 18C8 al 1869, 

 Milano, p. 167. 



