582 EEPOET— 1891. 



simple proportions is competent to produce tlie various kinds of symmetry 

 exliibited by crystals if the fundamental doctrine of Boscovich is admitted — that 

 the ultimate atoms are points endowed each with inertia, and with mutual attrac- 

 tions or repulsions dependent on mutual distances — repulsion manifesting itself at 

 the smallest distances and becoming infinite at infinitely small distances. 



After referring to the principal views which have been put forward as to the 

 nature of the molecules or units of crystals he goes on to argue that stable 

 equilibrium of a group of atoms endowed with Boscovich 's properties is evidently 

 found in that disposition of the atoms which gives the repulsions greatest play ; 

 that it is, in fact, the arrangement in which the packing is closest, or, in the lan- 

 guage of modern conceptions, the arrangement in which the potential energy of 

 the system is a minimum. 



He then proceeds to answer the question, AVhat groupmg of a concourse of 

 atoms will give closest packing ? first pointing out that the answer depends on 

 whether the atoms are of difl'erent kinds, and, if they are, on the numerical pro- 

 portion of each kind present, and also on the relative magnitude of the spaces 

 they occupy, or, in other words, on their relative capacities for repelling or being 

 repelled. 



For simplicity sake, he takes first the imaginary case of atoms confined to the 

 same plane, and points out that if there are two kinds of atoms present in equal 

 numbers, one of which exercises a feebler repulsion than the other, their repulsions 

 may be so proportioned that closest packing will be attained when one kind of 

 atom lies at the angles of a system of equal squares fitted close together, the other 

 at the centres of the same squares. 



lie then applies similar reasoning to cases of atoms not in the same plane, and, 

 after remarking that atoms which are all of one kind will pack closest when their 

 centres have the relative situation of the centres of a close-packed assemblage of 

 equal globes — a familiar example of which is found in the stacking of cannon- 

 shot — he states that the more general case of the closest packing of two or more 

 kinds of atoms is approximately depicted by the closest packing of globes, if the 

 globes are of different sizes, to represent the effects of the difference in the repul- 

 sions exercised by the different atoms. 



After saying that the nature of the grouping in which stable equilibrium is 

 found will depend on the ratio between the lengths of the radii of the globes 

 employed, the author traces the nature of the grouping for several particular 

 values of this ratio. 



He points out that not only hololiedral groupings corresponding to the simpler 

 forms of the crystallographic systems can be obtained in this way, but that the 

 more complicated partial symmetry of hemihedral and tetartohedral forms are 

 also to be obtained. 



As examples of the latter he gives a grouping in closest-packing that has the 

 precise symmetry of zinc-blende ZnS, which, according to Groth, crystallises in 

 the tetra'edrische hemiedrie of the cubic system, and another grouping that has the 

 precise symmetry of cuprite Cu-0, which, according to Groth, crystallises in the 

 plagiedrische hemiedrie of the cubic system. The numerical proportion of the 

 spheres of different radius employed is, in each case, that of the atoms present in 

 the molecule of the compound represented. 



Polar-pyroelectric phenomena and circular polarisation are, the author points 

 out, associated with peculiarities of the internal symmetry of the groupings, which 

 correspond in outward symmetry with tlie bodies displaying these phenomena. 



The grouping is portrayed by beads of different colours suspended in space 

 in the symmetrical manner requisite in each case. 



The author concludes his paper by referring to some geometrical properties of 

 the symmetrical systems of the crystallographer which he has discovered by an 

 extension of the methods adopted by Bravais and by Sohncke, and which have 

 greatly facilitated his work in finding symmetrical groupings to fit the forms and 

 composition of a variety of diflferent substances. 



