137 



mi lu r. the total solid volume of the replacing and replaced spheres 



nist be almost the same. Thus the cavity produced by three hydrogen- 

 itoms may be nearly filled up by one trivalent nitrogen-atom, etc. 

 It must lure be remarked that the proportionality of the volume 



f tin influence-spheres and the valencies of the respective atoms, 

 only an approximate one. This follows among other things from 



ic fact that the isomorphous substitution of K by Rb, or by Cs, 

 Iocs not lead to identical crystal-forms, and also from the fact that 



ic univalent Li-, or Afa-atoms do not replace those of K, Rb, Cs, 

 nnorphously. From this the possibility of a slight shifting of these 

 spheres in most closely packing after such substitutions, and, as a 



msequence of this, a change of symmetry, may be conceivable. 

 17. The theory just described in outlines, has been applied 



>y both the authors in a very ingenious way to explain a number 

 )f peculiarities and phenomena observed in crystals. For .these 

 ipplications we must refer the reader here to the series of publications 



icntioned in the preceding paragraph. It cannot be denied that 

 there often appears some arbitrariness with respect to the way in 

 which these assemblages are brought into agreement with the crys- 

 tallographical data at hand, especially with respect to the selection 

 )f the "multiples of the axial ratios" as calculated from direct mea- 

 surements. This is the same criticism which has often been actually 



lade of analogous trials made by other investigators to elucidate 

 relationships between crystalline forms of different, but closely 

 related substances. Multiplication of axial ratios with other numbers 

 than those following from the directly observed Millerian indices 

 )f the occurring crystal-facets, with the purpose of bringing out 

 analogies in form with the crystal-forms of other substances, is a 

 dangerous process. By suitable choice of the multipliers, all desired 

 axial ratios can finally be made comparable with each other. Not- 

 withstanding this, it can be seen from the B a r 1 o w-P o p e-theory 

 that a certain persistence of a particular type of structure as an 

 element throughout widely differing assemblages, often occurs in 

 the case of substances which are substitution-products of a same 

 mother-compound; and also, that the structures of two polymorphous 

 modifications of a same substance are often very simply related 

 to each other. 



It is of no use to study all the cases considered by the authors 

 in the light of these conceptions, nor to mention all the numerous 

 conclusions to which they have arrived, because, as we shall see 



