498 SCIENCE PROGRESS. 



all the particles to be identical, does implicitly recognise a 

 difference in their symmetry. Two molecules with their 

 respective environments which bear the relation to one 

 another of an object to its image, must differ completely 

 in their inherent symmetry, just as a right hand differs 

 from a left hand. For this reason Sohncke (19) accuses 

 Schonflies of constructing his crystal from two sorts of 

 molecules which bear this peculiar and arbitrary relation to 

 each other ; right- and left-handed molecules ; some struc- 

 tures consisting of right-handed, some of left-handed mole- 

 cules, and some of both. We may remark, however, that 

 the relation is not arbitrary, but results from the elementary 

 principles of symmetry of position ; further, that it harmon- 

 ises very well with the characters of optically active sub- 

 stances which crystallise in right- or left-handed forms 

 (dextro-chiral and laevo-chiral, as they are called by Lord 

 Kelvin) ; these appear to be really constructed of right- 

 and left-handed particles, for the character persists in the 

 solution. 



Here, however, we are beginning to travel outside the 

 pure geometry, and to raise questions belonging to the 

 physical aspect of the subject, which it is not the purpose 

 of the present article to pursue. 



Two remarks in conclusion. We have seen that the 

 thirty-two types can be deduced from two assumptions, as 

 was done by Hessel and Gadolin ; firstly, that the symmetry 

 of a crystal is homogeneous, i.e., that in any one crystal a 

 direction is symmetrically repeated in the same manner, 

 from whatever point that direction be taken ; secondly, that 

 the faces of a crystal obey the law of rational indices. 

 Now the first of these is a structural law, and is always 

 true, in a physical sense at least ; but the second is a law 

 relating to the external form of the crystal, and is a question 

 of superficial equilibrium between the crystal substance and 

 the medium from which it crystallises ; when a crystal is 

 immersed in a solution which is not concentrated, it assumes 

 a more or less rounded form, to which the law of rational 

 indices has no application. No doubt the law is a result of 

 the homogeneous structure, zmder certain conditions. 



