496 SCIENCE PROGRESS. 



any number of adjacent points ABC, etc. ; then, if we take 

 any other point P' of the system, it must be surrounded by 

 points A'B'C, etc., in such a manner that the lines PA, PB, 

 PC, etc., can be exactly superposed upon the lines P'A', 

 P'B', P'C, etc. Hence he derives all his structures by 

 considering in how many ways it is possible to move P to 

 P', and so on, by sliding, rotating and screw motions. 



Schonflies does not assume that the lines PA, PB, PC, 

 etc., can necessarily be only superposed upon P'A', P'B', 

 P'C, etc., but that PA = P'A', PB = P'B', PC = P'C, etc., 

 and that PA, PB, PC, etc., make the same angles with one 

 another, as P'A', P'B', P'C, etc. 



Now this does not exclude the possibility of one set of 

 lines bearing the same relation to the other that an object 

 does to its image in a mirror ; the two sets may be, in a 

 sense, identical, though not superposable or congruent. 

 Thus, for example, a point in the centre of a right hand 

 glove has precisely the same environment as a point in the 

 centre of a left hand glove, yet the two are not super- 

 posable ; the one is, as it were, the reflection of the other. 



In this way, then, Schonflies and Fedorow admit the 

 possibility of deriving one part of their system from another 

 by reflection ; in Sohncke's method such a reflection may 

 and does take place, but only as a result of the other 

 movements. 



Now the interesting feature of the new theory is that it 

 is a purely structural theory ; nothing is said about the 

 nature of the particles ; they may be identical throughout ; 

 and yet it accounts for the two essential features of a crystal, 

 the subjection of the faces to the law of rational indices, 

 and the existence of thirty-two types of symmetry. 



The law of rational indices is more or less implied in 

 the fact that all the 230 structures consist of interpenetrating 

 lattices (just as in Sohncke's theory), and, as has been 

 pointed out above, the planes of lattices obey this peculiar law. 

 In Sohncke's theory the law of rational indices is equally 

 implied for the same reason ; but the thirty-two types of 

 symmetry cannot be deduced without assuming that the 

 particles are of two or more different sorts. 



