composed of Rigid Particles in Contact. 471 



that, if the external spheres are fixed, the internal ones cannot 

 move, any distortion of the boundaries will cause an alteration 

 of the mean density, depending on the distortion and the 

 arrangement of the spheres. For example : — 



If arranged as a pile of shot (Plate X. fig. 2), which is an 

 arrangement of tetrahedra and octahedra, the density of the 



1 7T 



media is —i- ^, taking the density of the sphere as unity. 



If arranged in a cubical formation, as in fig. 1, the density 

 is t, or s/2 times less than in the former case. 



These arrangements are both controlled by the bounding 

 spheres ; and in either case the distortion necessitates a change 

 of volume. 



Either of these forms can be changed into the other by 

 changing the shape of the bounding surface. 



In both these cases the structure of the group is crystalline, 

 but that is on account of the plane boundaries. 



Practically, when the boundaries are not plane, or when 

 the grains are of various sizes or shapes, such media consist 

 of more or less crystalline groups having their axes in different 

 directions, so that their mean condition is amorphous. 



The dilation consequent on any distortion for a crystalline 

 group may be definitely expressed. When the mean condition 

 is amorphous, it becomes difficult to ascertain definitely what 

 the relations between distortion and dilation are. But if, 

 when at maximum density, the mean condition is not only 

 amorphous but isotropic, a natural assumption seems to be that 

 any small contraction from the condition of maximum density 

 in one direction means an equal extension in two others at 

 right angles. 



As such a contraction in one direction continues, the con- 

 dition of the medium ceases to be isotropic, and the relation 

 changes until dilation ceases. Then a minimum density is 

 reached ; after this, further contraction in the same direction 

 causes a contraction of volume, which continues until a 

 maximum density is reached. Such a relation between the 

 contraction in one direction and the consequent dilation would 

 be expressed by 



e— 1=^ia /sin 2 — ; 



e being the coefficient of dilation, a that of contraction, and 

 e x the maximum dilation ; the + ve root only to be taken. 



The amorphous condition of minimum volume is a very 

 stable condition ; but there would be a direct relation between 



2L2 



