330 BEPORT— 1901. 



The second part of the memoir considers the regular partitioning of a 

 plane, and of space, and the nature of zonohedra, or figures whose faces 

 intersect in parallel edges, and shows that there are six kinds of zono- 

 hedra. [Here also the author lays down the principles of simple elonga- 

 tion {Zug) and Shear ( Verschiebung), and shows that any parallelepiped 

 may be transformed into any other by these two processes. These 

 principles are chiefly of importance in Fedorow's development of his own 

 theory of crystal structure, and of his methods of calculation.] 



With regard to the partitioning of space, it is shown that space may 

 be filled either by equal figures ranged parallel to one another ; these are 

 called ' parallelohedra ' ; or by polyhedra, which, while equal or symmetri- 

 cally similar, are not necessarily parallel ; these are called ' stereohedra.' 



The plane-faced j^arallelohedra are bounded by pairs of parallel faces 

 (i.e., they possess centro-symmetry), and their arrangement is necessarily 

 that of a space lattice. There are four sorts of such parallelohedra, 

 namely, those with three, four, six, or seven pairs of parallel faces ; and 

 the filling of space with these corresponds to the close packing of spheres 

 which are in contact with six, eight, twelve, or eight neighbouring spheres 

 respectively. Fedorow's most general sort of parallelohedron, the fourth 

 of those mentioned above, the ' heptaparallelohedron,' is identical with 

 the ' tetrakaidekahedron ' subsequently and independently established by 

 Lord Kelvin as the most general parallel-faced cell into which space can 

 be regularly partitioned ; ^ its superficial area is, as was shown by both 

 authors, a minimum for a given volume. 



When space is partitioned into differently orientated identically 

 similar plane-faced stereohedra, these may always be grouped together into 

 sets, such that each set is a parallelohedron ; further, the analogous points 

 of the stereohedra constitute a regular point-system, just as the analogous 

 points of the parallelohedra constitute a space-lattice. Here, then, 

 we have a statement of the fact that the points of a regular point- 

 system can always be grouped into clusters whose arrangement is a space- 

 lattice. 



As will be seen hereafter, this conception of parallelohedra, as opposed 

 to stereohedra, forms the basis of Fedorow's own theory of crystal 

 structure. 



The last section of the memoir is occupied with the consideration of 

 polyliedra with concave faces, or ' koilohedra.' 



In his second treatise, that dealing with regular systems of figures,^ 

 the problem of crystal structure is more directly approached. A regular 

 system of figures is defined as consisting of an infinite assemblage of finite 

 figures, such that when any two of them are made to coincide by one of 

 the processes of symmetrical repetition (including herein the mirror-image 

 repetition to be mentioned presently) the whole system coincides with 

 itself again. This is, of course, practically the same as the definition of 

 Schonflies, and must lead to the same results. 



If any point in one of the figures be chosen, and the homologous 

 jioints in all the figures of the system be sought, the whole complex con- 

 stitutes a regular point-system. 



Those point-systems in which only repetition about axes (screw or 

 other) or simple translation is involved correspond to Sohncke's systems, 



1 Proc. Roy. Soc, 1894, Iv. p. 1. See also PJdl. Map., 1887, xsiv. p. 503. 

 " See for a short account ^eits. Kryst. Min,, 1802, vol. sx. pp. 39-62. 



