134 



assumption about the nature of the constituent motifs. As we have 

 seen ( 9), the solution of this problem involves the supposition of 

 two enantiomorphously related repeats, as soon as there is question 

 of patterns having symmetry-properties of the second order. 



The mathematical problem just mentioned has been solved by 

 Von Fedorow and by Schoenflies *); and although it would be 

 quite out of place here to give a full account of these deductions, 

 some general remarks as to the way followed by these authors may 

 be of interest. 



Both authors subdivide the unlimited space into an infinite number 

 of equal or enantiomorphously related, contiguous small volumes, 

 filling up that space completey. The distribution of matter within 

 such an elementary volume, - - which Von Fedorow calls a 

 sterohedron, while Schoenflies prefers the name of fundamental 

 domain for it, --is supposed to be completely arbitrary and free 

 from all symmetry. Its volume is constant and equal to that of 

 the elementary parallelepiped of the space-lattice, or a multiple 

 of it. When some of these identical or enantiomorphous "fundamental 

 domains" in symmetrical space-lattices or structures, are eventually 

 combined into greater units exhibiting a certain symmetry, these 

 symmetrical "complex domains", which by similar repetition are 

 also filling up the whole space, are discriminated by Von Fedorow 

 as parallelohedm 2 ). Corresponding points of such parallelohedra are 

 also corresponding (homologous) points of the regular system, and 



*) loco cit.; see also: A. Schoenflies, Zeits. f. Kryst., 54, 545, (1915); 55, 323, 

 (1916); F. Wallerant, Bull, de la Soc. Miner., 21, 197, (1898). 



2) If a point of a space-lattice be joined with all nearest points situated 

 round it, and planes perpendicular to the centre of these lines be constructed, 

 a volume of space is separated, which is limited by fourteen planes which 

 are pairwise parallel to each other. In a cubic space-lattice for instance, these 

 planes are perpendicular to the edges of the cubic cell and to the four cube- 

 diagonals. The "fundamental domain" thus determined, a hepta-parallelo- 

 hedron, is in the latter case a cube, the corners of which are truncated by 

 planes of the octahedron. With elements of this shape space can be filled 

 without any room remaining between the composing cells. These hepta-parrallelohe- 

 dra, already used by Lord Kelvin, have an important share in the 

 deductions of Von Fedorow. However, it may be remarked here, that it is 

 not necessary to determine the special shape of the fundamental domain. This 

 form can be quite arbitrary; but its volume is always constant and equal to 

 that of the elementary cell of the space-lattice, or in regular systems in 

 general a multiple of this. 



