Mr, "W. Barlow on Crystal Symmetry. 



19 



That the existence of these translations involves the pro- 

 perty that the mass is geometrically divisible into identical 

 portions which are similarly orientated may be shown in the 

 toll owing way : — 



Select some three directions in the structure, not in the 

 same plane, in which translations occur, and corresponding 

 to any point taken at random, locate by means of a continual 

 repetition of these translations a set of points identically 

 resembling the selected point ; the points thus formed will, it 

 is evident, form a parallelepipedal network. 



Surround each of the points of the network in a similar 

 manner with an identical cell,, sameway orientated, the cells 

 being small enough to avoid interpenetration, but otherwise 

 of an arbitrary form. Then let all the cells expand simultane- 

 ously and uniformly in every direction but only till the- 

 moving walls of contiguous cells meet, as in the partitioning 

 of space previously described * ; the result is to fill space with 

 the cells, and since, in the present case, the similarity of the 

 conditions of formation of each cell includes same way-orientation 

 these cells will, besides being all identical and containing 

 identical portions of the structure, be all same way-orientated. 

 The above proposition is therefore established. 



If a straight line be drawn through any two of the points 

 of a parallelepipedal network of identical points, it is easily 

 shown that this line passes through, an endless succession of 

 the points, and that if no other such point lies directly between 

 the two points, the distance between succeeding points on the 

 line is that separating the points referred to. Lines and planes 

 thus encountering a succession of equally spaced similar 

 points may, from this property, be called homogeneous lines or 

 homogeneous planes of points. The dotted lines ST and Y W 

 in figure 4 page 7 are homogeneous lines. 



The Law of Rational Indices Deduced. 



Proposition 16. — The identical sameway -orientated por- 

 tions of structure of the last proposition correspond to the 

 molecules soustractives of Haily and are available for the 

 use to which lie put the latter, which are indeed a specialized 

 form of the cells just described. The presence of the someway - 

 orientated portions of proposition 15 involves the existence of 

 the law of rational indices which connects the directions of 

 homogeneous planes and homogeneous lines of points. 



Proof. It has just been pointed out that points from 

 which the aspect of the structure is not only identically the 



* See p. 7. 

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