n34 HEPORt— 190L 



Schonflies, and must, as stated above, contain on their walls all the 

 elements of symmetry of the structure. 



The double system just obtained might have been constructed in another 

 way. Thus, draw a line from each point of the 24-punktner perpendi- 

 cular to a plane of symmetry of the structure, and produce it to an equal 

 distance on the opposite side ; the system so obtained is identical with 

 that previously obtained by the employment of centres of symmetry. 



A homogeneous structure consisting of material of any sort or shape 

 which contains points of two kinds like the above is identical with its 

 mirror-image. 



All the possible types of homogeneous structures are constructed in 

 the following way. Take a Sohncke point-system and, where possible, 

 insert into it the mirror-image of itself {i.e., the enantiomorphous Sohncke- 

 system) in such a way that the coincidence-movements of the two coincide 

 (e.g., the Sohncke-system obtained from E, of fig. 10, combined with that 

 obtained from L). 



The two constituent systems are then related to each other in one or 

 more of three modes, either (1) across a centre or centres of symmetry so 

 that they are oppositely orientated in every direction, or (2) across a plane 

 or planes of either ordinary or gliding symmetry, or (3) they are opposed 

 to each other with reference to one direction and are at the same time 

 orientated at right angles to each other. 



His third mode is in reality the method of repetition, used by 

 Fedorow, Schonflies, and Curie, which has an axis of alternating 

 symmetry ; Barlow employs it only in the case of the tetartohedral 

 symmetry of the tetragonal system because the other types which possess 

 symmetry of this nature possess also the symmetry of one of the other two 

 modes, and have therefore been already found. 



By applying these three modes of duplication to Sohncke's sixty-five 

 systems. Barlow deduced the same 230 types of symmetry which were 

 distinguished by Fedorow and Schonflies. 



The table of the 16.5 additional systems given by Barlow has this advan- 

 tage, that it distinguishes clearly the enantiomorphous systems from those 

 which possess mirror-image symmetry, and shows the mutual relations 

 of the two enantiomorphous systems of which a double system consists, 

 and further indicates the exact position of some of the centres and 

 planes of symmetry in the structure. 



Points of the structure which lie at these centres or upon axes or 

 planes of symmetry (' singular points ' of Barlow) are clearly less numerous 

 than any other sets of homologous points in the structure. In the stack 

 of cubes, for example, the centres of the cubes are less numerous than the 

 most general sorts of homologous points within the cubes. As explained 

 above, there are two sets of twenty-four points each surrounding each 

 centre ; it is evident, therefore, that the least symmetrically situated 

 points are no less than forty-eight time? as numerous as the centres. 



Barlow's theory of crystal structure, which is based upon the principle 

 of close-packing, will be considered later. 



Any account of the geometrical theories of crystal structure which 

 omitted reference to the important work of Lord Kelvin would be very 

 incomplete. This author has investigated the problem of the homo- 

 geneous partitioning of space, and, as was mentioned above, established 

 independently the tetrakaidekahedron (Fedorow'a heptaparallelohedron) 



