684 Dr. A. E. H. Tutton [March 14, 



possiljle faces to those which cut off small whole-number relat ve 

 lengths from the crystal axes. Indeed, it is because only those planes 

 which contain the points of the space-lattice are possible as crystal 

 faces that the law of rational indices obtains. For any three points 

 of the space-lattice determine a plane in which similar points are 

 analogously regularly repeated, and which is a possible crystal face 

 obeying the law of rational indices. Moreover, those facial planes 

 which are most densely strewn with points are of the greatest crystallo- 

 graphic importance, l)eing what are known as the primary faces, 

 either parallel to the crystal axes or cutting off unit lengths there- 

 from, as well as being usually the planes of cleavage. 



As the space-lattice units are all sameways orientated, any one 

 atom of the molecular or polymolecular grosser unit might be equally 

 well chosen as the representative point of the lattice, so long as a 

 similar choice were made in every space-lattice unit, and the resulting 

 space-lattice would be the same whichever atom vyere so selected. 

 Consequently, the space-lattice is afforded by the similarly (identically) 

 situated atoms of the same chemical element throughout the crystal 

 structure. The combined point-system (one of the 230 possible 

 point-systems) may thus be considered to be built up of as many 

 identical but interpenetrating space -lattices as there are atoms in the 

 space-lattice grosser unit. These facts are concisely expressed in the 

 definition of crystal structure which was stated as follows by Professor 

 yon Groth at the Cambridge meeting of the British Association in 1904. 



''A crystal — considered as indefinitely extended— consists of n 

 interpenetrating regular point-systems, each of which is formed from 

 similar atoms ; each of these point-systems is built up from n inter- 

 penetrating space-lattices, each of the latter being formed from atoms 

 occupying parallel positions. All the space-lattices of the combined 

 system are geometrically identical or are characterized by the same 

 elementary parallelepipedon." 



Having thus arrived at a comprehensive idea of crystal structure 

 on the assumption of each atom and each grosser space-lattice unit 

 being only a point, as far as which wc are on safe and assured ground, 

 we may proceed to the consideration of the various ideas advanced 

 concerning the character of the units of structure thus represented by 

 points, that is, concerning the mode in which the space around the 

 point is more or less filled up. 



The valency theory of Barlow and Pope considers the atomic 

 point to be expanded into the sphere of the atom's influence, the 

 relative size of which in any one substance is supposed to be propor- 

 tional to the fundamental valency of the chemical element of which 

 the atom is composed. The spheres are further assumed to be pressed 

 together on crystallization until they fill space, becoming thereby 

 deformed into polyhedra. The theory of von Fedorow, on the other 

 hand, considers the grosser or space-lattice units to be parallelohedra ; 

 besides those corresponding to the 14 space-lattices there are 9 other 



