TRANSACTION'S OF SECTION C. 635 



have whole numbers for the indices which fix its position, for along any line the 

 distance between two particles is by hypothesis a whole multiple of the common 

 distance between any two adjacent ones in the same line. Thus the first great 

 crystallographic law — the law of the rationality of the indices — is an immediate 

 consequence. 



In the next place, it was found that the possiblemodesof symmetry of arrange- 

 ment of the particles of such a system depend on the form of the parallelepiped, 

 and that any possible arrangement of the particles must present a symmetry which 

 is identical with one or other of the six completely symmetrical types already 

 referred to. And calculation shows that any other mode of grouping — a repetition 

 by fives or sevens, for example — round an axis of symmetry, would involve the pre- 

 sence of planes having irrational indices ; and this according to the first law is 

 impossible. 



The abeyance of symmetry, however, met with in the partially symmetrical 

 types required the aid of an auxiliary hypothesis — namely, that the abeyance of 

 symmetry belongs to the particle itself, and not to the arrangement of the 

 particles. 



But the parallelepipedal arrangement imagined by Bravais is unnecessarily 

 special. Our actual observations of physical characters relate not to single lines of 

 particles, but to groups of parallel lines of particles : the identity of character 

 observed in parallel directions is thus not necessarily due to actual identity of each 

 line with its neighbour, but may be due to statistical equality, an equality of 

 averages. If, for example, a plane were divided into regular hexagons, and a 

 particle were placed at each corner of each of these figures, the physical properties 

 of the system of particles would be the same along all lines parallel to each other 

 as far as experiment could decide, and yet the arrangement of the particles in the 

 plane, though possibly crystalline, is not that of a Bravais system. In any straight 

 line passing along the sides of a series of the hexagons, the particles will not be 

 equidistant from each other: they are in equidistant pairs, and the two nearest 

 particles of adjacent pairs are twice as far from each other as the particles of the 

 same pair. 



Sohncke accordingly suggested a more general definition than that of Bravais 

 for the regularity of the arrangement, a definition which had been proposed some 

 years before by Wiener — namely, that the grouping relative to any one particle is 

 identical with that relative to any other. This definition admits of the possibility 

 of the hexagonal arrangement j ust mentioned ; further, it allows of the orientation 

 of the particles themselves being diflferent in adjacent lines. Following a mathe- 

 matical process which had been already employed by Jordan, Sohncke deduced all 

 the possible modes of grouping consistent with the new definition, and for a time 

 was under the impression that the types of symmetry found by him to be mathe- 

 matically possible are exactly identical with those already referred to ; and this 

 without introducing the auxiliary hypothesis relative to partial symmetry of the 

 elementary particles of merosymmetrical crystals, except in cases of hemimorphism. 

 It was, however, pointed out by Wulfl^, who has himself made valuable contri- 

 butions to the subject, that though no unknown crystallographic type belongs to 

 such a regular arrangement, one type of symmetry, that presented by dioptase, is 

 missing ; and it seems that, in this case at least, the merosymmetry can only be 

 accounted for by the merosymmetry of the particle, or something equivalent to 

 it, if the definition of regularity suggested by Sohncke is to be accepted. It was 

 recognised by Sohncke that each of his point-systems can be regarded as a com- 

 posite Bravais system, one of the latter being repeated in various positions corre- 

 sponding with the symmetry of the parallelepiped itself. 



More recently, Schonflies has made a more general hypothesis still — namely, 

 that in each substance, whether its crystals be completely or partially symmetrical 

 in facial development, the particles are not of a single kind, but of two kinds, 

 related to each other in form in much the same way as a right-hand glove and a 

 left-hand glove. With this hypothesis he finds that all the thirty-two known types 

 are accounted for without any specialisation of the characters of the particle, and 

 that no other type of symmetry is mathematically possible. 



