10 Mr. W. Barlow on Crystal Symmetry. 



Therefore, if any two space-units A, B be selected, some 

 screw-spiral movement can be made of the entire mass which 

 will transfer the unit A to the exact spot initially occupied 

 by B and leave the aspect of the ultimate structure as viewed 

 from any fixed point precisely the same after as before the 

 change. The screw-spiral movement may be resolved into 

 two components, a rotation and a translation : the former 

 only is concerned in effecting any change of orientation 

 requisite in bringing A to the place of B, and any right line 

 drawn to intersect A to the direction of an identically corre- 

 sponding line passing through B. If the aspect of the 

 structure regarded from the centre of A has the same 

 orientation as its aspect viewed from the centre of B, the 

 rotation component of the movement will be zero, and this 

 movement will then be a translation. 



A very simple illustration may make matters clearer : — 

 Suppose space to be divided symmetrically into cells which 

 are identical triangular prisms ; half of the prisms then have 

 the opposite orientation to that of the other half. If it is 

 desired to bring a cell to the precise place of some other 

 cell, a movement which is a translation will suffice when the 

 two cells are saraeways orientated, but a screw movement or 

 a rotation will be requisite when the two cells have not the 

 same orientation. Any two of the identically placed space- 

 units of a homogeneous structure can be selected for the 

 application of a coincidence-movement of the nature de- 

 scribed, and this movement may take place in either direction, 

 i. e. from A to B or from B to A. All such coincidence-move- 

 ments as are possible for the given structure, when they are 

 taken collectively, constitute, where the structure is supposed 

 of infinite extent, the infinite group of related movements 

 referred to. The change of orientation effected by the rota- 

 tion component of any one of these movements is adequately 

 defined if the direction of its axis and the amount and direction 

 of the rotation are given. 



Pkoposition 3. — In order to locate in any given homogeneous 

 structure the various directions vjhich are identical to a given 

 direction, it suffices to apply to the latter in succession one of 

 each of the different rotations ivhich enter as components into 

 those movements of the group of movements referred to which 

 are applicable to some single space-unit. Thus in the trigonal 

 symmetrical pattern of fig. 4, there are two directions identical 

 to the direction of A B, viz. those of C D and E F, derived 

 by rotation through 120° and 240° respectively. 



Peoof. As already shown, if two directions bear a similar 

 relation to the structure, any line drawn in one of them to 



