a78 



NA TURE 



\yan. 17, 1889 



Finally, parallel with these, place a series of similar 

 equidistant webs in such positions that the points in succes- 

 sive planes lie at equal intervals upon straight lines whose 

 direction (o Bj) is determined by the points in the first 

 two webs. 



In this way a network of points is constructed, in which 

 the line joining any two points is a thread, and the plane 

 through any three points is a web. 



The space inclosed by six adjacent planes of the system 

 having no other points of the network between them is a 

 parallelopiped (o Aj \\ c,), from which the whole system 

 may be constructed by repetition, and which may be taken 

 to represent the structural element {molecule soustractive) 

 of Haiiy. 



The complete investigation of all possible solid networks 

 led Bravais to the conclusion that these, if classified by 

 the character of their symmetry, fall into seven groups, 

 which correspond exactly to the seven systems into 

 which crj'stals are grouped in accordance with their 

 symmetry. 



It follows, then, that two (not, however, independent) 

 features of crystals are fully accounted for by a parallelo- 

 pipedal arrangement of points in space — namely, the 

 ■symmetry of the crystallographic systems and the law 

 which governs the inclinations of the faces (law of rational 

 indices). 



FiG.l. 



Thereare, however, subdivisions of the various systems 

 consisting of the merohedral or partially symmetrical 

 crystals belonging to them, which are not explained by the 

 geometry of a network ; these consequently were referred 

 by Bravais, not merely to the arrangement of the molecules 

 in space, but also to the internal symmetry of the molecule 

 itself 



Hence the theory of Bravais, while able to a certain 

 extent to explain the form of crystals, requires an auxiliary 

 hypothesis if it is to explain those modifications which are 

 partially symmetrical or merohedral. 



Sohncke, treating the problem in a different manner, 

 and reasoning from the fact that the properties of a crystal 

 are the same at any one point within its mass as at any 

 other but different along different directions, inquired in 

 how many ways a system of points may be arranged in 

 space so that the configuration of the system round any 

 one point is precisely similar to that round any other. 

 S,uch a configuration may be called a Sohncke system 

 of points in space {i-egelmiissiges Piinktsysteiii). 



From his analysis of this problem, it appears that there 

 ^ire sixty-five possible Sohncke systems of points, and that 

 these may be grouped according to their symmetry into 

 seven classes corresponding to the seven crystallographic 

 systems ; and further that there are withm each class 



minor subdivisions, characterized by a partial symmetry 

 corresponding to the hemihedral and tetartohedral form; 

 of crystallographers. 



It may be expected, then, that the theory of Sohncke 

 contains within itself the essential features of a Bravais 

 network of structural molecules, and also the auxiliary 

 hypothesis regarding the arrangement of parts within the 

 molecule which is required to account for merohedrism. 



Now, on closer examination the arrangement of Sohncke 

 does prove to be a simple extension of that of Bravais. 



Each of Sohncke's arrangements may in fact be regarded 

 as derived from one of the parallelopipedal networks of 

 Bravais if for every point of the latter be substituted a 

 group of symmetrically arranged satellites. It is not 

 necessary that any particle in a group of these satellites 

 should actually coincide with the point of the Bravais 

 network from which the group is derived ; and the points 

 of the Sohncke system do not themselves form a net- 

 work ; it is only -when all the points in each group of 

 satellites are condensed into one centre that a Sohncke 

 system coincides with a Bravais network. 



To any particle of one of the satellite groups corre- 

 sponds in every other group a particle similarly situated 

 with regard to the point from which the group has been 

 derived. Every such point may be said to be homologous 

 with the first. 



It will then be found that each complete set of homo- 

 logous points is itself a Bravais network in space, and that 

 consequently any Sohncke system may be regarded as a 

 certain number of congruent networks interpenetrating 



F10.2. 



one another : the number of such networks is in general 

 equal to the number of points which constitute each group 

 of satellites. 



The relation of a Sohncke system to the network from 

 which it is derived may be illustrated by a bees'-cell 

 distribution of points in one plane, i.e. by points which 

 occupy the angles of a series of regular hexagons. Thus 

 in the adjoining figure the dots form a Sohncke system 

 in one i)lane, smce the configuration of the system round 

 any one point is similar to that round any other ; but they 

 do not form a Bravais web, since the points do not lie 

 at equal distances along straight lines. 



If, however, points, represented in the figure by the 

 circles o, be placed at the centres of the hexagons, they 

 will by themselves constitute a web, and the hexagonal 

 system may be derived from this web by replacing 

 each of its points by a group of two satellites, A and B. 

 Or, from the second point of view, the arrangement 

 may be regarded as a triangular web. A, completely 

 interpenetrated by a similar web, B. 



It is a remarkable feature of the Sohncke systems that 

 some among them are characterized by a spiral disposition 

 of the particles along the threads of a right- or left- 

 handed screw : now this spiral character, which does not 

 belong to any of the Bravais networks, supplies a geo- 

 metrical basis for the right- or left-handed nature of some 

 merohedral crystals which possess the property of right- or 

 left-handed rotatory polarization. 



The theory of Sohncke as sketched above appeared to 



