8 Mr. W. Barlow on Crystal Symmetry. 



The cells obtained in this way may be called space-units*. 

 A plane example of units of this nature is given by fig. 4 ; 

 a symmetrical pattern, which is a plane homogeneous struc- 

 ture of the nature defined, being shown divided up into such 

 units by dotted lines. 



Before stating the other important property of homogene- 

 ous structures, two propositions, based on the one just put 

 forward, may be given. 



Pkoposition 1. — Each of the cells, or space-units, comprises 

 one of each of the differently -placed mathematical points of the 

 structure and only one ; except, however, the points formir ^ 

 the boundary which are common to two or more cells, and 

 all, or nearly all of which, occur twice in each cellf. 



Pkoof. For, by hypothesis, the cells are as numerous as 

 any set of identical mathematical points in the structure, and 

 therefore any kind of position cannot occur more than once 

 within any particular cell; since, if there were two points 

 within one cell identical, there would be two such points in 

 every cell, making twice as many as the number of cells. 



Proposition 2. — The maximum, number of different direc- 

 tions which can be identical with one another in any given 

 structure, is that of the different orientations of its space-units ; 

 in other words, that of the different orientations of the mass 

 which can be made without altering the aspect of its structure 

 as viewed in any fixed direction. 



Proof. Let a line in some given direction, selected at 

 random, te drawn to intersect a certain space-unit. Then, 

 since according to proposition 1, no two points within this 

 unit are alike, no other line can be drawn to cut the unit in 

 the same manner. 



But if any different directions in the mass are identically 

 alike, every right line drawn in one of them must have 

 corresponding to it as many differently-directed right lines 

 cutting the ultimate structure in identically the same manner 

 as there are other directions identical to the given direction. 

 And therefore, if one of these similarly situated lines cuts 

 one of the space-units in a certain manner, every other of 



* Schonflies calls tliem " einfache Fundamentalbereiche," and regards 

 their properties as attaching- to the molecules of which the structure 

 consists. (See Krystallsysteme mid Krystallstructuv, von Dr. Arthur 

 Schonflies, Leipzig, 1891, pp. 572 & 616.) The} r are not, however, in any 

 sense actual or physical molecules. Their shape, as appears from the 

 method of obtaining them just given, is generally more or less arbitrary. 



t See Krystallsysteme und Krystallstructur, p. 572. The points in the 

 boundaries which do not occur twice are what the writer has called in 

 another place sinyular points. (See Zeitschr. fur Kryst. xxiii. p. 60.) 

 Such points are marked P, Q, R in fig. 4. 



